STAT 3021 Chapter4.pdf - STAT 3021 Spring 2018 Chapter 4...

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STAT 3021 Spring 2018 Chapter 4. Mathematical Expectation 4.1 Mean of a Random Variable In mathematical expectation, the expected value is calculated by using the probability distribution. We shall refer the expected value as the mean of the random variable X or the mean of te probability distribution of X and write it as μ X or simply μ when it it clear to which the random variable we refer. The mean of a random variable is usually understood as a “center” value of the underlying probability distribution. Definition 4.1 Example 1. (Example 4.3 on page 114) Let X be the random variable that denotes the life in hours of a certain electronic device. The probability density function is f ( x ) = 20000 x 3 , x > 100 , 0 , elsewhere . Find the expected life of this type of device. 1
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STAT 3021 Spring 2018 Next, we generalize the expected value to random variable related to X , or say g ( X ), a function of X . Theorem 4.1 Example 2. (Example 4.4 on page 115) Suppose that the number of cars X that pass through a car wash between 4:00 P.M. and 5:00 P.M. on any sunny Friday has the following probability distribution: x 4 5 6 7 8 9 f ( x ) 1/12 1/12 1/4 1/4 1/6 1/6 Let g ( X ) = 2 X - 1 represent the amount of money, in dollars, paid to the attendant by the manager. Find the attendant’s expected earnings for this particular time period. We shall now extend our concept of mathematical expectation to the case of two random variables X and Y with joint probability distribution f ( x, y ). Definition 4.2 2
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STAT 3021 Spring 2018 Example 3. (Exercise 4.23 on page 118) Suppose that X and Y have the following joint probability function: f ( x, y ) x 2 4 y 1 0.10 0.15 3 0.20 0.30 5 0.10 0.15 Find the expected value of g ( X, Y ) = XY 2 . Example 4. (Exercise 4.26 on page 118) Let X and Y be random variables with joint density function f ( x ) = 4 xy, 0 < x, y < 1 , 0 , elsewhere . Find the expected value of Z = X 2 + Y 2 . 3
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STAT 3021 Spring 2018 4.2 Variance and Covariance of Random Variables The variance is a measure of the dispersion, or variability in the distribution. These two measures are useful summaries of the probability distribution of X.
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