Discrete Probability Distributions: A Comprehensive Guide for Students, Study notes of Probability and Statistics

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CHAPTER 6

DISCRETE

PROBABILITY

DISTRIBUTIONS

By: Wandi Ding

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6.1 DISCRETE RANDOM VARIABLES

Review: A variable was defined as a characteristic or attribute

that can assume different values.

Let’s see two examples below:

1) If a fair die is rolled, let X indicates the outcomes, and X

could be 1, 2, 3, 4, 5, or 6 by chance

2) If two coins are tossed, let Y indicates the number of heads,

and Y could be 0,1 or 2 by chance.

Like variables X,Y called random variable.

Random variable: is a variable whose values are determined by

chance.

Discrete variable: values can be counted (may have finite or

infinite values), like above examples, X and Y are discrete

variables.

Continuous variable: can assume all values in the interval

between any two given values.

Ex: the weight for everybody in our classroom.

the height for everybody in our classroom. 2

6.1 DISCRETE RANDOM VARIABLES

Analyze: X could be 0, 1, 2 , or 3. If we put all possible values and there corresponding probabilities in a table, we will have the below:

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Number of boys X

Probability P(x)

Tables like this are called probability distribution. Probability distribution : consists of all the possible values for a random variable and corresponding probability of the value. The probabilities are determined by calssical method or empirical method. Probability histogram: is a histogram in which the horizontal axis indicates the values of a random variable and vertical axis indicates the corresponding probability.

Let’s see the probability distribution for the above example in next slide:

6.1 DISCRETE RANDOM VARIABLES

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0 1 2 3 More

probability P(x)

X

Proper of a probability distribution:

1. The sum of the probability of all possible values in the sample space

must equal to 1; that is

1. The probability of each value in the sample space must be between 0

and 1, that is 0 ≤P(x)≤

 P ( x ) 1

6.1 DISCRETE RANDOM VARIABLES

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Variance for a discrete random variable:

these two formulas can be used either one.

standard deviation of a discrete random:

Ex: The below table shows a probability distribution for the

random variable X which represents the number of shots made for a basketball player to shoot three free throws, X could be 0,1,2, or 3.

2 2 2 2

 X  [( x  X ] * P ( x )] [ x * P ( x )]  X

 (^) x   x

X 0 1 2 3

P(x) 0.01 0.10 0.38 0.

6.1 DISCRETE RANDOM VARIABLES

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Q:calculate the mean, variance and standard deviation for

the discrete variable.

Ans: Mean

=00.01+10.1+20.38+30.51=2.

 X  x * p ( x )

x P(x) 0 0.01 (0-2.39)^2 *0.01=0. 1 0.1 (1-2.39)^2 *0.1=0. 2 0.38 (2-2.39)^2 *0.38=0. 3 0.51 (3-2.39)^2 *0.51=0.

( x   X )* p ( x )

( x   X )^2 * p ( x ) 0. 4979

The variance is

The standard deviation is

( ) * ( ) 0. 4979 0. 5

 X   x   X P x  

 (^) X  ^2  0. 4979  0. 7

6.2 BINOMIAL PROBABILITY DISTRIBUTION

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Binominal probability distribution function (PDF):

The probability of obtaining x successes in n independent

trials of a binomial experiment is given by:

where p is the probability of success.

Mean(expected value) and standard deviation of a

binomial distribution:

A binominal experiment with n independent trials and

probability of success P has a mean and standard deviation given by the formulas:

P x C p p x n x n x ( ) (^) n x ( 1  )  0 , 1 , 2 ,..., 

 X  np and  X  np ( 1  p )

6.2 BINOMIAL PROBABILITY DISTRIBUTION

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Example 3, 4 shows a good application on binominal

probability

Example 6 shows how to use binominal table to find the

probability

Example 7 shows how to create a histogram for a

binominal distribution.

Let’s take more time to learn these four examples.