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
Example 4
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x y 6
Independent Experiments 7
Example 8
Generalization 9
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Example 11
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Bernoulli trial 17
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Example 20
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