Tutorial on Discrete Probability Distributions.

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  • Topic: Random variable, Discrete probability distribution, Probability distribution
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Tutorial on Discrete Probability Distributions
Tutorial on discrete probability distributions with examples and detailed solutions.

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| Let X be a random variable that takes the numerical values X1, X2, ..., Xn with probablities p(X1), p(X2), ..., p(Xn) respectively. A discrete probability distribution consists of the values of the random variable X and their corresponding probabilities P(X). 

The probabilities P(X) are such that ∑ P(X) = 1Example 1:Let the random variable X represents the number of boys in a family. 

a) Construct the probability distribution for a family of two children. 

b) Find the mean and standard deviation of X.Solution to Example 1:  * a) We first construct a tree diagram to represent all possible distributions of boys and girls in the family.  * Assuming that all the above possibilities are equally likely, the probabilities are: 

P(X=2) = P(BB) = 1 / 4 
P(X=1) = P(BG) + P(GB) = 1 / 4 + 1 / 4 = 1 / 2 
P(X=0) = P(GG) = 1 / 4  * The discrete probability distribution of X is given by  X| P(X)| 0| 1 / 4|
1| 1 / 2|
2| 1 / 4|
* * Note that ∑ P(X) = 1 * b) The mean µ of the random variable X is defined by µ = ∑ X P(X)

= 0 * (1/4) + 1 * (1/2) + 2 * (1/4) = 1  * The standard deviation σ of the random variable X is defined by σ = Square Root [ ∑ (X- µ) 2 P(X) ]

= Square Root [ (0 - 1) 2 * (1/4) + (1 - 1) 2 * (1/2) + (2 - 1) 2 * (1/4) ] 

= 1 / square root (2)Example 2:Two balanced dice are rolled. Let X be the sum of the two dice. 

a) Obtain the probability distribution of X. 

b) Find the mean and standard deviation of X.Solution to Example 2:  * a) When the two balanced dice are rolled, there are 36 equally likely possible outcomes as shown below .  * The possible values of X are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.  * The possible outcomes are equally...
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