The document discusses discrete and continuous probability distributions, explaining that a discrete distribution applies to variables that can take on countable values while a continuous distribution is used for variables that can take any value within a range. It provides examples of discrete variables like coin flips and continuous variables like weights. The document also outlines the differences between discrete and continuous probability distributions in how they are represented and calculated.
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.
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.
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).
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
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).
It includes various cases and practice problems related to Binomial, Poisson & Normal Distributions. Detailed information on where tp use which probability.
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
This slide presentation is a non-technical introduction to the concept of probability. The level of the presentation would be most suitable for college students majoring in business or a related field, but it could also be used in high school classes.
Mathematical Background for Artificial Intelligenceananth
Mathematical background is essential for understanding and developing AI and Machine Learning applications. In this presentation we give a brief tutorial that encompasses basic probability theory, distributions, mixture models, anomaly detection, graphical representations such as Bayesian Networks, etc.
This presentation covers important topics such as
Multiple Independent Random Variables or i.i.d samples.
Expectations or Expected values
T-Distribution
Central Limit Theorem
Asymptotics & Law of Large Numbers
Confidence Intervals
BINOMIAL ,POISSON AND NORMAL DISTRIBUTION.pptxletbestrong
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
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
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
3. • Probability is a measure of the expectation that an
event will occur or a statement is true.
• Probabilities are given a value between 0 (will not
occur) and 1 (will occur).
• The higher the probability of an event, the more
certain we are that the event will occur.
4. • The concept has been given an axiomatic
mathematical derivation in probability theory,
which is used widely in such areas of study as
mathematics, statistics, finance, gambling, science,
artificial intelligence/machine learning and
philosophy to, for example, draw inferences about
the expected frequency of events.
• Probability theory is also used to describe the
underlying mechanics and regularities of complex
systems.
5. Etymology
• The word Probability derives from the Latin
probabilitas, which can also mean probity, a
measure of the authority of a witness in a legal case
in Europe, and often correlated with the witness's
nobility.
• In a sense, this differs much from the modern
meaning of probability, which, in contrast, is a
measure of the weight of empirical evidence, and is
arrived at from inductive reasoning and statistical
inference.
6. Interpretations
• When dealing with experiments that are
random and well-defined in a purely
theoretical setting (like tossing a fair coin),
probabilities describe the statistical number of
outcomes considered divided by the number
of all outcomes (tossing a fair coin twice will
yield HH with probability 1/4, because the
four outcomes HH, HT, TH and TT are
possible).
7. • When it comes to practical application,
however, the word probability does not have a
singular direct definition.
• There are two major categories of probability
interpretations, whose adherents possess
conflicting views about the fundamental
nature of probability:
8. • Objectivists assign numbers to describe some
objective or physical state of affairs.
• The most popular version of objective probability is
frequentist probability, which claims that the
probability of a random event denotes the relative
frequency of occurrence of an experiment's
outcome, when repeating the experiment.
• This interpretation considers probability to be the
relative frequency "in the long run" of outcomes.
• A modification of this is propensity probability,
which interprets probability as the tendency of
some experiment to yield a certain outcome, even if
it is performed only once.
9. • Subjectivists assign numbers per subjective
probability, i.e., as a degree of belief.
• The most popular version of subjective probability
is Bayesian probability, which includes expert
knowledge as well as experimental data to produce
probabilities.
• The expert knowledge is represented by some
(subjective) prior probability distribution. The data
is incorporated in a likelihood function.
• The product of the prior and the likelihood,
normalized, results in a posterior probability
distribution that incorporates all the information
known to date.
10. Probability Distributions:
Discrete vs. Continuous
• All probability distributions can be classified as
discrete probability distributions or as
continuous probability distributions,
depending on whether they define
probabilities associated with discrete variables
or continuous variables
11. Discrete vs. Continuous Variables
• If a variable can take on any value between
two specified values, it is called a continuous
variable; otherwise, it is called a discrete
variable
12. • Suppose the fire department mandates that all fire
fighters must weigh between 150 and 250 pounds.
The weight of a fire fighter would be an example of
a continuous variable; since a fire fighter's weight
could take on any value between 150 and 250
pounds.
• Suppose we flip a coin and count the number of
heads. The number of heads could be any integer
value between 0 and plus infinity. However, it could
not be any number between 0 and plus infinity. We
could not, for example, get 2.5 heads. Therefore,
the number of heads must be a discrete variable.
14. Discrete Probability Distributions
• DEFINITION:
If a random variable is a discrete variable, its
probability distribution is called a discrete
probability distribution.
15. EXAMPLE
• Suppose you flip a coin two times. This simple
statistical experiment can have four possible
outcomes: HH, HT, TH, and TT.
• Now, let the random variable X represent the
number of Heads that result from this experiment
• The random variable X can only take on the values
0, 1, or 2, so it is a discrete random variable.
16. • The probability distribution for this statistical
experiment appears below
• The above table represents a discrete probability
distribution because it relates each value of a discrete
random variable with its probability of occurrence
Number of
heads
Probability
0 0.25
1 0.50
2 0.25
17. Continuous Probability Distributions
• DEFINITION:
If a random variable is a continuous variable,
its probability distribution is called a
continuous probability distribution.
18. • A continuous probability distribution differs from a
discrete probability distribution in several ways:
1. The probability that a continuous random variable
will assume a particular value is zero.
2.As a result, a continuous probability distribution
cannot be expressed in tabular form.
3.Instead, an equation or formula is used to describe
a continuous probability distribution.
19. • Most often, the equation used to describe a
continuous probability distribution is called a
probability density function.
• Sometimes, it is referred to as a density function, a
PDF, or a pdf. For a continuous probability
distribution, the density function has the following
properties:
20. • Since the continuous random variable is defined
over a continuous range of values (called the
domain of the variable), the graph of the density
function will also be continuous over that range.
• The area bounded by the curve of the density
function and the x-axis is equal to 1, when
computed over the domain of the variable.
• The probability that a random variable assumes a
value between a and b is equal to the area under
the density function bounded by a and b.
21. • For example, consider the probability density
function shown in the graph below:
Suppose we wanted to know the probability that
the random variable X was less than or equal to a.
The probability that X is less than or equal to a is
equal to the area under the curve bounded by a
and minus infinity - as indicated by the shaded area.
23. • Note: The shaded area in the graph represents
the probability that the random variable X is
less than or equal to a. This is a cumulative
probability. However, the probability that X is
exactly equal to a would be zero. A continuous
random variable can take on an infinite
number of values. The probability that it will
equal a specific value (such as a) is always
zero.