Hw7Sol - SIEO 3600 (IEOR Majors) Introduction to...

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SIEO 3600 (IEOR Majors) Assignment #7 Introduction to Probability and Statistics March 9, 2010 Assignment #7 – due March 9th, 2010 ( Expectation, variance and covariance of random variables ) 1. Ten balls are randomly chosen from an urn containing 17 white and 23 black balls. Let X denote the number of white balls chosen. (a) If everytime we put the chosen ball back to the urn ( with replacement ), compute E [ X ] by defining appropriate indicator variables X i , i = 1 ,..., 10 so that X = 10 X i =1 X i . (b) If everytime we don’t put the ball back ( without replacement ), compute E [ X ] by defining appropriate indicator variables Y i , i = 1 ,..., 17 so that X = 17 X i =1 Y i . Solution: (a) For i = 1 ,..., 10, let X i = ± 1 if the i th pick is a white ball, 0 otherwise. With replacement, everything we have the same number of white and black balls. Hence, E ( X i ) = P ( X i = 1) = 17 / 40. Linearity of expectation implies that E ( X ) = 10 i =1 E ( X i ) = 10 · E ( X i ) = 17 / 4. (b) With all 17 white balls labeled with j = 1 ,..., 10, let Y j = ± 1 if white ball number j is picked, 0 otherwise. Without replacement, this is equivalent to picking 10 balls out of 40 at one time. Note that E ( Y j ) = P ( Y j = 1) = P (white ball number j is picked) = total number of ways to pick 10 balls out of 40 with white ball number j included total number of ways to pick 10 balls out of 40 = ( 39 9 ) ( 40 10 ) = 1 4 . Hence, linearity of expectation implies that E ( X ) = 10 j =1 E ( Y j ) = 17 · E ( Y j ) = 17 / 4. Although the polices in (a) and (b) are different (with or with replacement), the answers are the same. (Unbelievable!)
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2 SIEO 3600, Assignment #7 2. Let X be a continuous random variable with CDF F and PDF f , then define its median as the value m such that F ( m ) = 1 / 2. (Think about the sample median of a data set). The median , like the mean, is important in predicting the value of a random variable. It can be shown that the mean E ( x ) is the best predictor from the point of view of minimizing the expected value of the square of the error. Prove that the median is the best predictor if one wants to minimize the expected value of the absolution error . That is, E [ | X - c | ] is minimized when c is the median of X . Hint: Write E [ | X - c | ] = Z -∞ | x - c | f ( x ) dx = Z c -∞ | x - c | f ( x ) dx + Z c | x - c | f ( x ) dx = Z c -∞ ( c - x ) f ( x ) dx + Z c ( x - c ) f ( x ) dx = cF ( c ) - Z c -∞ xf ( x ) dx + Z c xf ( x ) dx - c [1 - F ( c )] , and then choose the best c to minimize E [ | X - c | ]. Solutions
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Hw7Sol - SIEO 3600 (IEOR Majors) Introduction to...

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