Random Variables and Stochastic Processes – Dr. Ghazi Al Sukkar Office Hours: will be posted soon Course Website: 1 Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul /

2 Chapter 3 Repeated Trials Combined Experiments Bernoulli Trials

Combined Experiments 3

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Independent Experiments 7

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Generalization 9

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Bernoulli trial 17

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Chapter 4 Distribution Functions and Random Variables  Random Variables  Distribution Functions  Discrete Random Variables  Expectations of Discrete Random Variables  Variances and Moments of Discrete Random Variables  Standardized Random Variables 21

Random Variables 22 S

Definition I 23

Definition II

Examples Example ： If in rolling two fair dice, X is the sum, then X can only assume the values 2, 3, 4, …, 12 with the following probabilities ： P(X=2) = P({(1,1)}) = P(X=3) = P({(1,2), (2,1)}) = P(X=4) = P({(1,3), (2,2), (3,1)}) = etc.. 25

Example: Suppose that 3 cards are drawn from an ordinary deck of 52 cards, one by one, at random and with replacement. Let X be the number of spades drawn; then X is a random variable. If an outcome of spades is denoted by s, and other outcomes are represented by t, then X is a real-valued function defined on the sample space S={(s,s,s), (t,s,s), (s,t,s), (s,s,t), (t,t,s), (t,s,t), (s,t,t), (t,t,t)}  X(s,s,s) = 3, X(t,s,s) = X(s,s,t) = X(s,t,s) = 2, What are the probabilities of X = 0, 1, 2, 3 ? Sol ： 26

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Example: In the United States, the number of twin births is approximately 1 in 90. Let X be the number of births in a certain hospital until the first twins are born. X is a random variable. Denote twin births by T and single births by N. Then X is a real-valued function defined on the sample space The set of all possible values of X is {1, 2, 3, …} 28

Ans: 3/5 29