BINOMIAL DISTRIBUTION
In probability theory and statistics, the binomial distribution is the discrete probability distribution gives only two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.
Number of trials (n) is a fixed number.
The outcome of a given trial is either success or failure.
The probability of success (p) remains constant from trial to trial which means an experiment is conducted under homogeneous conditions.
The trials are independent which means the outcome of previous trial does not affect the outcome of the next trial.
Binomial Probability Distribution
In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution.
There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙.
The binomial distribution formula is for any random variable X, given by;
P(x:n,p) = nCx px (1-p)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x
Binomial Distribution Mean and Variance
For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas
Mean, μ = np
Variance, σ2 = npq
Standard Deviation σ= √(npq)
Where p is the probability of success
q is the probability of failure, where q = 1-p
Properties of binomial distribution
The properties of the binomial distribution are:
• There are two possible outcomes: true or false, success or failure, yes or no.
• There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
• The probability of success or failure remains the same for each trial.
• Only the number of success is calculated out of n independent trials.
• Every trial is an independent trial, which means the outcome of one trial does not affect the outcome
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
Random Variable
Discrete Probability Distribution
continuous Probability Distribution
Probability Mass Function
Probability Density Function
Expected value
variance
Binomial Distribution
poisson distribution
normal distribution
The ppt gives an idea about basic concept of Estimation. point and interval. Properties of good estimate is also covered. Confidence interval for single means, difference between two means, proportion and difference of two proportion for different sample sizes are included along with case studies.
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. Suppose we flip a coin two times and count the number of heads (successes).
A Probability Distribution is a way to shape the sample data to make predictions and draw conclusions about an entire population. It refers to the frequency at which some events or experiments occur. It helps finding all the possible values a random variable can take between the minimum and maximum statistically possible values.
Random Variable
Discrete Probability Distribution
continuous Probability Distribution
Probability Mass Function
Probability Density Function
Expected value
variance
Binomial Distribution
poisson distribution
normal distribution
The ppt gives an idea about basic concept of Estimation. point and interval. Properties of good estimate is also covered. Confidence interval for single means, difference between two means, proportion and difference of two proportion for different sample sizes are included along with case studies.
A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution. Suppose we flip a coin two times and count the number of heads (successes).
The PPT covered the distinguish between discrete and continuous distribution. Detailed explanation of the types of discrete distributions such as binomial distribution, Poisson distribution & Hyper-geometric distribution.
Elements of Inference covers the following concepts and takes off right from where we left off in the previous slide https://www.slideshare.net/GiridharChandrasekar1/statistics1-the-basics-of-statistics.
Population Vs Sample (Measures)
Probability
Random Variables
Probability Distributions
Statistical Inference – The Concept
Probability and Some Special Discrete DistributionsDoyelGhosh1
I HAVE DISCUSSED THEORY OF PROBABILITY IN AXIOMATIC APPROACH AS WE KNOW THERE WERE MANY DRAWBACKS OF CLASSICAL APPROACH. I HAVE ALSO DISCUSSED ABOUT SOME DISCRETE PROBABILITY DISTRIBUTIONS.
Multilateralism is a process of organizing relations between groups of three or more states in pursuit of a common goal. Multilateralism is based on certain principles that shape the character of the arrangement or institution, such as cooperation, equality, and legitimacy. Multilateralism often favors strengthening the United Nations and other international institutions that involve as many of the world's nations as possible.
Research reporting is the oral or written presentation of the findings in such detail and forms as to be readily understood and assessed by the society , economy or particularly by the researchers.
Report writing is common to both academic and managerial situations . In academics ,a research report is prepared for comprehensive and application oriented learning . In businesses or organizations reports used for the basis of decision making .
Footnotes are conventional procedures used in scholarly writing in validate or to explain certain aspects in the main text . Such devices should be used sparingly and only when the material being presented clearly needs amplification or acknowledgement . Foot note should appear only in the body of a paper or thesis never in an abstract. Footnotes can be distracting if they are so numerous and frequent that they persistently impinge upon the readers attention . Therefore, it becomes essential, before including any footnotes in a paper or essay , to asses whether the material being relegated to a foot note is important enough to incorporated into the main body of the text , or whether it is essential to include it at all.
The one-sample t-test is used to determine whether a sample comes from a population with a specific mean. This population mean is not always known, but is sometimes hypothesized.
Preparation and Presentation of budget.pptxletbestrong
Budget is the annual financial statement of a government which lays out fiscal roadmap for the country for the next one year. It is prepared by the ministry of finance in consultation with Niti Aayog and other concerned ministries.Budget is the annual financial statement of a government which lays out fiscal roadmap for the country for the next one year. It is prepared by the ministry of finance in consultation with Niti Aayog and other concerned ministries.The Union Budget of India, referred to as the annual Financial Statement in Article 112 of the Constitution of India, is the annual budget of the Republic of India, presented each year on the last working day of February by the Finance Minister of India in Parliament. The budget has to be passed by the House before it can come into effect on April 1, the start of India's financial year.The origins of the modern Budget can be traced to the Norman period, where two departments dealt with finance the Treasury and the Exchequer. The Treasury received and paid out money on behalf of the monarch. The Exchequer, had a 'lower office' which received money, and an 'upper office', concerned with regulating the Kings accountsThe term budget has been derived from the old French word bougette, which means a leather bag or wallet. The first use of the term 'budget' may date back to 1733 financial statement by Walpole as Prime Minister and Chancellor of the Exchequer. A cartoon of him opening a patent medicine seller's wares was published at the time, as a satirical comment with the caption 'The Budget Opened'. ('Budge' is an old word for a bag or small case). Initially, budget referred solely to the Chancellor’s annual speech on the nations finances. Now, the term is used for an annual financial statement of income and expenditure of a government.
Human capital refers to the stock of skill, ability, expertise, education, and knowledge in a nation at a point of time. We need investment in human capital to produce more human capital out of human resources.Nations require adequate human capital who are educated and qualified as educators and other specialists. In other words, we need great human capital to create other human capital like doctors, engineers, professors, etc., which will later become a human asset and contribute to the economy of the country.Human resources are the people who are part of the workforce and contribute to the productivity of a country. The quality and efficiency of human resources depend on factors such as health, education, skills, and motivation. Different countries have different levels of human resource development and potential. For example, India has a large and young population that can provide a demographic dividend if properly educated and employedThe term human resources refers to the size of the population of a country along with its efficiency, educational qualities, productivity, organisational abilities and farsightedness. It is the ultimate resource, but not equally distributed over the worldIndia has 62.5% of its population in the age group of 15-59 years which is ever increasing and will be at the peak around 2036 when it will reach approximately 65%.These population parameters indicate an availability of demographic dividend in India, which started in 2005-06 and will last till 2055-56.According to Economic Survey 2018-19,India’s Demographic Dividend will peak around 2041, when the share of working-age,i.e. 20-59 years, population is expected to hit 59%.India has one of the youngest populations in an aging world. By 2020, the median age in India will be just 28, compared to 37 in China and the US, 45 in Western Europe, and 49 in Japan.Since 2018, India’s working-age population (people between 15 and 64 years of age) has grown larger than the dependents population — children aged 14 or below as well as people above 65 years of age. This bulge in the working-age population is going to last till 2055, or 37 years from its beginning.This transition happens largely because of a decrease in the total fertility rate(TFR, which is the number of births per woman) after the increase in life expectancy gets stabilised.A study on demographic dividend in India by United Nations Population Fund (UNFPA) throws up two interesting facts.The window of demographic dividend opportunity in India is available for five decades from 2005-06 to 2055-56, longer than any other country in the world.This demographic dividend window is available at different times in different states because of differential behaviour of the population parameter.
Monopsony in labour market is a situation in which there is only one firm to buy the services of a particular type of labour. Hence it is regarded as a “buyer’s monopoly”. The term monopsony is derived from the Greek words: mono which means ‘one and posinia which means ‘a buying’.
Monopolistic situations occur when the labour market is imperfect. There is immobility of labour-both occupational and geographical. This is because labour in a particular area is of a special type. It is trained for a particular type of work and its services cannot be utilised by any other firm except the one for which it is specialised. There may be certain other forces preventing labour to migrate to other areas.
Micro finance institutions :
Micro finance institution (MFI) are financial companies that provides small loans to people who do not have any access to banking facilities . The definition of small loans varies between different countries . In India ,all loans that are below Rs. 1 lakh can be considered as a microloans .
Although most microfinance institutions target the eradication of poverty as their motive , some of the new entrants are focussed on the sale of more products to consumers .
Goals of microfinance institutions
Transform into a financial institution that assists in the development of communities that are sustainable .
Help in the provision of resources that offer support to the lower sections of the society .There is a special focus on women in this regard ,as they have emerged successful in setting up income generation enterprise .
Evaluate the options available to help eradicate poverty at a faster rate .
Mobilise self employment opportunities.
AS PER WORLD BANK DATA , CLOSE TO 1.7 BILLION PEOPLE ACROSS MULTIPLE COUNTRIES DO NOT HAVE ACCESS TO BASIC FINACIAL SERVICES . THIS IS WHERE MICROFINANCE INSTITUTIONS PLAY A MAJOR ROLE .
Key benefits
It enables people expand their present opportunities .
It provides easy access to credit facilities
It make future investments possible
It serves the under –financed section of the society
It helps in the generation of employment opportunity
It inculcates the discipline of saving
It brings about significant economic gains
It results in better credit management practices
It results in better education
Microfinance includes the following products:
Microloans - Microfinance loans are significant as these are provided to borrowers with no collateral. The end result of microloans should be to have its recipients outgrow smaller loans and be ready for traditional bank loans.
Microsaving’s – Microsaving’s accounts allow entrepreneurs operate savings accounts with no minimum balance. These accounts help users inculcate financial discipline and develop an interest in saving for the future.
Microinsurance - Microinsurance is a type of coverage provided to borrowers of microloans. These insurance plans have lower premiums than traditional insurance policies.
In some situations, recipients of microloans are expected to take some training courses, such as cash flow management or book-keeping.
Groups Organised by Micro finance Institutions in India
Joint Liability Group (JLG )
This is usually a informal group that consists of 4-10 individuals who seek loans against mutual guarantee .Each individual in a JLG is equally responsible for the loan repayment in a timely manner .
Self Help Group
It is a group of individual with similar socio- economic backgrounds .These small entrepreneurs come together for a short duration and create a common fund for their business needs .
Evolution of population policy
Radha kamal Mukherjee Committee (1940)
Bhore Committee(1943)
India became one of the first developing countries to come up with a state – sponsored family Planning programme in the 1950
In 1952 a population policy committee was established .
Foreign Direct Investment
Foreign direct investment is a financial investment made by a company based in another country that owns a controlling stake in a company in another country .
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
Yes of course, you can easily start mining pi network coin today and sell to legit pi vendors in the United States.
Here the telegram contact of my personal vendor.
@Pi_vendor_247
#pi network #pi coins #legit #passive income
#US
Currently pi network is not tradable on binance or any other exchange because we are still in the enclosed mainnet.
Right now the only way to sell pi coins is by trading with a verified merchant.
What is a pi merchant?
A pi merchant is someone verified by pi network team and allowed to barter pi coins for goods and services.
Since pi network is not doing any pre-sale The only way exchanges like binance/huobi or crypto whales can get pi is by buying from miners. And a merchant stands in between the exchanges and the miners.
I will leave the telegram contact of my personal pi merchant. I and my friends has traded more than 6000pi coins successfully
Tele-gram
@Pi_vendor_247
Seminar: Gender Board Diversity through Ownership NetworksGRAPE
Seminar on gender diversity spillovers through ownership networks at FAME|GRAPE. Presenting novel research. Studies in economics and management using econometrics methods.
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
where can I find a legit pi merchant onlineDOT TECH
Yes. This is very easy what you need is a recommendation from someone who has successfully traded pi coins before with a merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi network coins and resell them to Investors looking forward to hold thousands of pi coins before the open mainnet.
I will leave the telegram contact of my personal pi merchant to trade with
@Pi_vendor_247
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
The Evolution of Non-Banking Financial Companies (NBFCs) in India: Challenges...beulahfernandes8
Role in Financial System
NBFCs are critical in bridging the financial inclusion gap.
They provide specialized financial services that cater to segments often neglected by traditional banks.
Economic Impact
NBFCs contribute significantly to India's GDP.
They support sectors like micro, small, and medium enterprises (MSMEs), housing finance, and personal loans.
Lecture slide titled Fraud Risk Mitigation, Webinar Lecture Delivered at the Society for West African Internal Audit Practitioners (SWAIAP) on Wednesday, November 8, 2023.
2. BINOMIAL DISTRIBUTION
In probability theory and statistics, the binomial distribution is the discrete probability distribution gives only
two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be
only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass
or fail. This distribution is also called a binomial probability distribution.
Number of trials (n) is a fixed number.
The outcome of a given trial is either success or failure.
The probability of success (p) remains constant from trial to trial which means an experiment is conducted
under homogeneous conditions.
The trials are independent which means the outcome of previous trial does not affect the outcome of the next
trial.
3. Binomial Probability Distribution
In binomial probability distribution, the number of ‘Success’ in a sequence of n
experiments, where each time a question is asked for yes-no, then the valued
outcome is represented either with success/yes/true/one (probability p) or
failure/no/false/zero (probability q = 1 − p). A single success/failure test is also
called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a
Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a
Bernoulli distribution.
4. There are two parameters n and p used here in a binomial distribution. The
variable ‘n’ states the number of times the experiment runs and the variable
‘p’ tells the probability of any one outcome. Suppose a die is thrown
randomly 10 times, then the probability of getting 2 for anyone throw is ⅙.
When you throw the dice 10 times, you have a binomial distribution of n =
10 and p = ⅙.
6. The binomial distribution formula is for any random variable X, given by;
P(x:n,p) = nCx px (1-p)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials,
where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x
7. Binomial Distribution Mean and Variance
For a binomial distribution, the mean, variance and standard deviation for the given number of success
are represented using the formulas
Mean, μ = np
Variance, σ2 = npq
Standard Deviation σ= √(npq)
Where p is the probability of success
q is the probability of failure, where q = 1-p
8. PROPERTIES OF BINOMIAL DISTRIBUTION
The properties of the binomial distribution are:
There are two possible outcomes: true or false, success or failure, yes or no.
There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
The probability of success or failure remains the same for each trial.
Only the number of success is calculated out of n independent trials.
Every trial is an independent trial, which means the outcome of one trial does not
affect the outcome of another trial.
9. It is a p.m.f.(probability mass function).
The parameters of Binomial Distribution are ’n’ and ‘p’ where, n - number of trials and p - probability of
success.
Mean of Binomial Distribution = np
Variance of Binomial Distribution = npq (here q=1-p)
Mode of Binomial Distribution is integral part of (n+1)p, if (n+1)p is not an integer. But, if (n+1)p is an
integer, then the distribution has two modal values, (n+1)p and [(n+1)p] - 1.
q- probability of failure, q < 1, hence, npq < np. Thus, Variance is less than Mean.
If p=q=1/2, then the distribution is symmetric about median and if p is not equal to q, then it is skewed
distribution.
Additive property of Binomial Distribution : If X and Y are independent variables such that X follows
Binomial Distribution with (n1, p) and Y follows Binomial Distribution with (n2, p), then (X+Y) follows
Binomial Distribution with (n1+n2, p).
If X1, X2,…., Xn are independent and identically distributed Bernoulli variates each with parameter p, then
their addition follows Binomial Distribution with (n, p).
10. What are the criteria for the binomial
distribution
• The number of trials should be fixed.
• Each trial should be independent.
• The probability of success is exactly the same from one trial to
the other trial.
11. Examples of binomial distribution problems:
The number of defective/non-defective products in a production run.
Yes/No Survey (such as asking 150 people if they watch ABC news).
Vote counts for a candidate in an election.
The number of successful sales calls.
The number of male/female workers in a company
So, as we have the basis let’s see some binominal distribution examples, problems, and solutions from
real life.
12. Binomial Distribution Vs Normal Distribution
• Binomial Distribution Vs Normal Distribution
• The main difference between the binomial distribution and the
normal distribution is that binomial distribution is discrete,
whereas the normal distribution is continuous. It means that the
binomial distribution has a finite amount of events, whereas the
normal distribution has an infinite number of events. In case, if
the sample size for the binomial distribution is very large, then
the distribution curve for the binomial distribution is similar to the
normal distribution curve
•
14. Poisson Distribution
• Poisson distribution is a theoretical discrete probability and is also known
as the Poisson distribution probability mass function. It is used to find the
probability of an independent event that is occurring in a fixed interval of
time and has a constant mean rate. The Poisson distribution probability
mass function can also be used in other fixed intervals such as volume,
area, distance, etc. A Poisson random variable will relatively describe a
phenomenon if there are few successes over many trials. The Poisson
distribution is used as a limiting case of the binomial distribution when the
trials are large indefinitely. If a Poisson distribution models the same
binomial phenomenon, λ is replaced by np. Poisson distribution is named
after the French mathematician Denis Poisson
15. Poisson distribution
• Poisson distribution definition is used to model a discrete
probability of an event where independent events are occurring in
a fixed interval of time and have a known constant mean rate. In
other words, Poisson distribution is used to estimate how many
times an event is likely to occur within the given period of time. λ
is the Poisson rate parameter that indicates the expected value of
the average number of events in the fixed time interval. Poisson
distribution has wide use in the fields of business as well as in
biology
16. Example :
• Let us try and understand this with an example, customer care
center receives 100 calls per hour, 8 hours a day. As we can see
that the calls are independent of each other. The probability of
the number of calls per minute has a Poisson probability
distribution. There can be any number of calls per minute
irrespective of the number of calls received in the previous
minute. Below is the curve of the probabilities for a fixed value of
λ of a function following Poisson distribution:
17. FORMULA
• Poisson distribution formula is used to find the probability of an
event that happens independently, discretely over a fixed time
period, when the mean rate of occurrence is constant over time.
The Poisson distribution formula is applied when there is a large
number of possible outcomes. For a random discrete variable X
that follows the Poisson distribution, and λ is the average rate of
value, then the probability of x is given by:
• f(x) = P(X=x) = (e-λ λx )/x!
•
18. Where
x = 0, 1, 2, 3...
e is the Euler's number(e = 2.718)
λ is an average rate of the expected value and λ = variance, also λ>0
Poisson Distribution Mean and Variance
For Poisson distribution, which has λ as the average rate, for a fixed interval of
time, then the mean of the Poisson distribution and the value of variance will
be the same. So for X following Poisson distribution, we can say that λ is the
mean as well as the variance of the distribution.
19. Hence: E(X) = V(X) = λ
where
E(X) is the expected mean
V(X) is the variance
λ > 0
Properties of Poisson Distribution
The Poisson distribution is applicable in events that have a large number of rare and independent
possible events
20. Important Notes
• The formula for Poisson distribution is f(x) = P(X=x) = (e-λ λx )/x!.
• For the Poisson distribution, λ is always greater than 0.
• For Poisson distribution, the mean and the variance of the
distribution are equal.
21. PROPERTIES
• The events are independent.
• The average number of successes in the given period of time alone can occur.
No two events can occur at the same time.
• The Poisson distribution is limited when the number of trials n is indefinitely
large.
• mean = variance = λ
• np = λ is finite, where λ is constant.
• The standard deviation is always equal to the square root of the mean μ.
• The exact probability that the random variable X with mean μ =a is given by
P(X= a) = μa / a! e -μ
• If the mean is large, then the Poisson distribution is approximately a normal
distribution
22. Poisson distribution table
• Similar to the binomial distribution, we can have a Poisson
distribution table which will help us to quickly find the probability
mass function of an event that follows the Poisson distribution.
The Poisson distribution table shows different values of Poisson
distribution for various values of λ, where λ>0. Here in the table
given below, we can see that, for P(X =0) and λ = 0.5, the value of
the probability mass function is 0.6065 or 60.65%.
26. What are the properties of the normal distribution?
• The normal distribution is a continuous probability distribution that is
symmetrical on both sides of the mean, so the right side of the centre is
a mirror image of the left side.
• The area under the normal distribution curve represents probability and
the total area under the curve sums to one.
• Most of the continuous data values in a normal distribution tend to
cluster around the mean, and the further a value is from the mean, the
less likely it is to occur. The tails are asymptotic, which means that they
approach but never quite meet the horizon (i.e. x-axis).
• For a perfectly normal distribution the mean, median and mode will be
the same value, visually represented by the peak of the curve
27. The normal distribution is often called the bell curve because
the graph of its probability density looks like a bell. It is also
known as called Gaussian distribution, after the German
mathematician Carl Gauss who first described it.
28. What is the difference between a normal distribution and a
standard normal distribution?
A normal distribution is determined by two parameters the mean and the variance. A normal distribution with a
mean of 0 and a standard deviation of 1 is called a standard normal distribution.
29. Why is the normal distribution important?
The bell-shaped curve is a common feature of nature and psychology
The normal distribution is the most important probability distribution
in statistics because many continuous data in nature and psychology displays
this bell-shaped curve when compiled and graphed.
For example, if we randomly sampled 100 individuals we would expect
to see a normal distribution frequency curve for many continuous variables,
such as IQ, height, weight and blood pressure.
30. Parametric significance tests require a normal distribution of the samples' data points
The most powerful (parametric) statistical tests used by psychologists require
data to be normally distributed. If the data does not resemble a bell curve researchers may
have to use a less powerful type of statistical test, called non-parametric statistics.
Converting the raw scores of a normal distribution to z-scores
We can standardized the values (raw scores) of a normal distribution by
converting them into z-scores.
This procedure allows researchers to determine the proportion of the values
that fall within a specified number of standard deviations from the mean (i.e. calculate the
empirical rule).
31. What is the empirical rule formula?
• The empirical rule in statistics allows researchers to determine
the proportion of values that fall within certain distances from the
mean. The empirical rule is often referred to as the three-sigma
rule or the 68-95-99.7 rule.
32.
33. If the data values in a normal distribution are converted to standard score (z-score)
in a standard normal distribution the empirical rule describes the percentage of the
data that fall within specific numbers of standard deviations (σ) from the mean (μ)
for bell-shaped curves.
The empirical rule allows researchers to calculate the probability of randomly
obtaining a score from a normal distribution.
68% of data falls within the first standard deviation from the mean. This means there
is a 68% probability of randomly selecting a score between -1 and +1 standard
deviations from the mean.
34. 95% of the values fall within two standard deviations from the
mean. This means there is a 95% probability of randomly selecting
a score between -2 and +2 standard deviations from the mean.
99.7% of data will fall within three standard deviations from the
mean. This means there is a 99.7% probability of randomly
selecting a score between -3 and +3 standard deviations from the
mean.