Lecture Notes on Probability

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Unformatted text preview: re independent if p(X = r1 and Y = r2) = p(X = r1)· p(Y = r2). Theorem 5: If X and Y are independent variables on a sample space S, then E(XY) = E(X)E(Y). see text for the proof 26 Variance Deviation: The deviation of X at s S is X(s) E(X), the difference between the value of X and the mean of X. Definition 4: Let X be a random variable on the sample space S. The variance of X, denoted by V(X) is That is V(X) is the weighted average of the square of the deviation of X. The standard deviation of X, denoted by σ(X) is defined to be Theorem 6: If X is a2 random variable on a sample space S, then V(X) = E(X2) E(X) . Corollary 1: If X is a random variable on a sample space S and E(X) = µ , then V(X) = E((X µ)2). 27 Variance Example: What is the variance of the random variable X, where X(t) = 1 if a Bernoulli trial is a success and X(t) = 0 if it is a failure, where p is the probability of success and q is the probability of failure? Solution: Because X takes only the values 0 and 1, it follows that X2(t) = X(t). Hence, V(X) = E(X2) E(X)2 = p p2 = p(1 p) = pq. Variance of the Value of a Die: What is the variance of a random variable X, where X is the number that comes up when a fair die is rolled? Solution: We have V(X) = E(X2) E(X)2 . In an earlier example, we saw that E(X) = 7/2. Note that E(X2) = 1/6(12 + 22 + 32 +42 + 52 + 62) =...
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This document was uploaded on 03/29/2014 for the course COT 3100h at University of Central Florida.

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