The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
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Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
Random Variable (Discrete and Continuous)Cess011697
Learning Competencies
- to recall statistical experiment and sample space
- to illustrate a random variable (discrete and continuous).
- to distinguish between a discrete and a continuous random variable.
Chapter 4 part3- Means and Variances of Random Variablesnszakir
Statistics, study of probability, The Mean of a Random Variable, The Variance of a Random Variable, Rules for Means and Variances, The Law of Large Numbers,
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.4: The Central Limit Theorem
Random Variable (Discrete and Continuous)Cess011697
Learning Competencies
- to recall statistical experiment and sample space
- to illustrate a random variable (discrete and continuous).
- to distinguish between a discrete and a continuous random variable.
Random variables and probability distributions Random Va.docxcatheryncouper
Random variables and probability distributions
Random Variable
The outcome of an experiment need not be a number, for example, the outcome when a
coin is tossed can be 'heads' or 'tails'. However, we often want to represent outcomes
as numbers. A random variable is a function that associates a unique numerical value
with every outcome of an experiment. The value of the random variable will vary from
trial to trial as the experiment is repeated.
There are two types of random variable - discrete and continuous.
A random variable has either an associated probability distribution (discrete random
variable) or probability density function (continuous random variable).
Examples
1. A coin is tossed ten times. The random variable X is the number of tails that are
noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.
2. A light bulb is burned until it burns out. The random variable Y is its lifetime in
hours. Y can take any positive real value, so Y is a continuous random variable.
Expected Value
The expected value (or population mean) of a random variable indicates its average or
central value. It is a useful summary value (a number) of the variable's distribution.
Stating the expected value gives a general impression of the behaviour of some random
variable without giving full details of its probability distribution (if it is discrete) or its
probability density function (if it is continuous).
Two random variables with the same expected value can have very different
distributions. There are other useful descriptive measures which affect the shape of the
distribution, for example variance.
The expected value of a random variable X is symbolised by E(X) or µ.
If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and p(xi)
denotes P(X = xi), then the expected value of X is defined by:
where the elements are summed over all values of the random variable X.
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#discvar
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#contvar
http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#variance
If X is a continuous random variable with probability density function f(x), then the
expected value of X is defined by:
Example
Discrete case : When a die is thrown, each of the possible faces 1, 2, 3, 4, 5, 6 (the xi's)
has a probability of 1/6 (the p(xi)'s) of showing. The expected value of the face showing
is therefore:
µ = E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6) = 3.5
Notice that, in this case, E(X) is 3.5, which is not a possible value of X.
See also sample mean.
Variance
The (population) variance of a random variable is a non-negative number which gives
an idea of how widely spread the values of the random variable are likely to be; the
larger the variance, the more scattered the obser ...
The Normal Distribution:
There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
Steps for calculating probability using the Z-
score:
-Sketch a bell-shaped curve,
- Shade the area (which represents the probability)
-Use the Z-score formula to calculate Z-value(s)
-Look up Z-values in table
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Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Mrbml004 : Introduction to Information Theory for Machine LearningJaouad Dabounou
La quatrième séance de lecture de livres en machine learning.
Vidéo : https://youtu.be/Ab5RvD7ieFg
Elle concernera une brève introduction à la théorie de l'information: Entropy, K-L divergence, mutual Information,... et son application dans la fonction de perte et notamment la cross-entropy.
Lecture de trois livres, dans le cadre de "Monday reading books on machine learning".
Le premier livre, qui constituera le fil conducteur de toute l'action :
Christopher Bishop; Pattern Recognition and Machine Learning, Springer-Verlag New York Inc, 2006
Seront utilisées des parties de deux livres, surtout du livre :
Ian Goodfellow, Yoshua Bengio, Aaron Courville; Deep Learning, The MIT Press, 2016
et du livre :
Ovidiu Calin; Deep Learning Architectures: A Mathematical Approach, Springer, 2020
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
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2. Section 6.1
Discrete & Continuous Random Variables
After this section, you should be able to…
APPLY the concept of discrete random variables to a
variety of statistical settings
CALCULATE and INTERPRET the mean (expected value) of
a discrete random variable
CALCULATE and INTERPRET the standard deviation (and
variance) of a discrete random variable
DESCRIBE continuous random variables
3. Random
Variables
• A random variable, usually
written as X, is a variable
whose possible values are
numerical outcomes of a
random phenomenon.
• There are two types of
random variables, discrete
random variables and
continuous random
variables.
4. Discrete Random Variables
• A discrete random variable is one which may take on
only a countable number of distinct values such as 0,
1, 2, 3, 4,....
• Discrete random variables are usually (but not
necessarily) counts.
• Examples:
• number of children in a family
• the Friday night attendance at a cinema
• the number of patients a doctor sees in one day
• the number of defective light bulbs in a box of ten
• the number of “heads” flipped in 3 trials
5. Consider tossing a fair coin 3 times.
Define X = the number of heads obtained
X = 0: TTT
X = 1: HTT THT TTH
X = 2: HHT HTH THH
X = 3: HHH
Value 0 1 2 3
Probability 1/8 3/8 3/8 1/8
The probability distribution of a discrete random variable is
a list of probabilities associated with each of its possible
values
Probability Distribution
6. If you roll a pair of dice and record the sum for each trial, after
an infinite number of times, your distribution will approximate
the probability distribution on the next slide.
Rolling Dice: Probability
Distribution
7.
8. Discrete Random Variables
A discrete random variable X takes a fixed set of possible
values with gaps between. The probability distribution of
a discrete random variable X lists the values xi and their
probabilities pi:
Value: x1 x2 x3 …
Probability: p1 p2 p3 …
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. The sum of the probabilities is 1.
To find the probability of any event, add the probabilities pi
of the particular values xi that make up the event.
9. Describing the (Probability) Distribution
When analyzing discrete random variables, we’ll follow the
same strategy we used with quantitative data – describe the
shape, center (mean), and spread (standard deviation), and
identify any outliers.
10. Example: Babies’ Health at Birth
Background details are on page 343.
(a)Show that the probability distribution for X is legitimate.
(b)Make a histogram of the probability distribution. Describe
what you see.
(c)Apgar scores of 7 or higher indicate a healthy baby. What is
P(X ≥ 7)?
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
11. Example: Babies’ Health at Birth
Background details are on page 343.
(a)Show that the probability distribution for X is legitimate.
(b)Make a histogram of the probability distribution. Describe
the distribution.
(c)Apgar scores of 7 or higher indicate a healthy baby. What is
P(X ≥ 7)?
(a) All probabilities are between 0 and 1 and the probabilities
sum to 1. This is a legitimate probability distribution.
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
12. Example: Babies’ Health at Birth
b. Make a histogram of the probability distribution. Describe what you see.
c. Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)?
(b) The left-skewed shape of the distribution suggests a randomly
selected newborn will have an Apgar score at the high end of the scale.
While the range is from 0 to 10, there is a VERY small chance of getting
a baby with a score of 5 or lower. There are no obvious outliers. The
center of the distribution is approximately 8.
(c) P(X ≥ 7) = .908
We’d have a 91 % chance
of randomly choosing a
healthy baby.
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
13. Mean of a Discrete Random Variable
The mean of any discrete random variable is an average of
the possible outcomes, with each outcome weighted by its
probability.
Suppose that X is a discrete random variable whose
probability distribution is
Value: x1 x2 x3 …
Probability: p1 p2 p3 …
To find the mean (expected value) of X, multiply each
possible value by its probability, then add all the products:
x E(X) x1p1 x2 p2 x3 p3 ...
xi pi
14. Example: Apgar Scores – What’s Typical?
Consider the random variable X = Apgar Score
Compute the mean of the random variable X and interpret it in context.
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
x E(X) xi pi
15. Example: Apgar Scores – What’s Typical?
Consider the random variable X = Apgar Score
Compute the mean of the random variable X and interpret it in context.
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
x E(X) xi pi
(0)(0.001)(1)(0.006)(2)(0.007)...(10)(0.053)
8.128
The mean Apgar score of a randomly selected newborn is 8.128.
This is the long-term average Agar score of many, many
randomly chosen babies.
Note: The expected value does not need to be a possible value of
X or an integer! It is a long-term average over many repetitions.
16. Standard Deviation of a Discrete
Random Variable
The definition of the variance of a random variable is similar to
the definition of the variance for a set of quantitative data. To
get the standard deviation of a random variable, take the square
root of the variance.
Suppose that X is a discrete random variable whose
probability distribution is
Value: x1 x2 x3 …
Probability: p1 p2 p3 …
and that µX is the mean of X. The variance of X is
Var(X) X
2
(x1 X )2
p1 (x2 X )2
p2 (x3 X )2
p3 ...
(xi X )2
pi
17. Example: Apgar Scores – How Variable Are They?
Consider the random variable X = Apgar Score
Compute the standard deviation of the random variable X and
interpret it in context.
Value: 0 1 2 3 4 5 6 7 8 9 10
Probability: 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
X
2
(xi X )2
pi
(08.128)2
(0.001)(18.128)2
(0.006)...(108.128)2
(0.053)
2.066
The standard deviation of X is 1.437. On average, a randomly
selected baby’s Apgar score will differ from the mean 8.128 by
about 1.4 units.
X 2.066 1.437
Variance
18. Continuous Random Variable
• A continuous random variable is one which
takes an infinite number of possible values.
• Continuous random variables are usually
measurements.
• Examples:
– height
– weight
– the amount of sugar in an orange
– the time required to run a mile.
19. Continuous Random Variables
A continuous random variable X takes on all values in an
interval of numbers. The probability distribution of X is
described by a density curve. The probability of any
event is the area under the density curve and above the
values of X that make up the event.
20. • A continuous random variable is
not defined at specific values.
• Instead, it is defined over an
interval of value; however, you
can calculate the probability of a
range of values.
• It is very similar to z-scores and
normal distribution calculations.
Continuous Random Variables
21. Example: Young Women’s Heights
The height of young women can be defined as a continuous
random variable (Y) with a probability distribution is N(64,
2.7).
A. What is the probability that a randomly chosen young
woman has height between 68 and 70 inches?
P(68 ≤ Y ≤ 70) = ???
22. Example: Young Women’s Heights
The height of young women can be defined as a continuous random variable (Y) with a
probability distribution is N(64, 2.7).
A. What is the probability that a randomly chosen young woman has height between
68 and 70 inches?
P(68 ≤ Y ≤ 70) = ???
z
6864
2.7
1.48
z
7064
2.7
2.22
P(1.48 ≤ Z ≤ 2.22) = P(Z ≤ 2.22) – P(Z ≤ 1.48)
= 0.9868 – 0.9306
= 0.0562
There is about a 5.6% chance that a randomly chosen young
woman has a height between 68 and 70 inches.
23. Example: Young Women’s Heights
The height of young women can be defined as a continuous
random variable (Y) with a probability distribution is N(64,
2.7).
B. At 70 inches tall, is Mrs. Daniel unusually tall?
24. Example: Young Women’s Heights
The height of young women can be defined as a continuous random variable (Y) with a
probability distribution is N(64, 2.7).
B. At 70 inches tall, is Mrs. Daniel unusually tall?
P(Y ≤ 70) = ???
z
7064
2.7
2.22
P value: 0.9868
Yes, Mrs. Daniel is unusually tall because 98.68% of the
population is shorter than her.