20notes4.pdf - Notes 20 Desired outcomes from last class...

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Notes 20 Desired outcomes from last class Students will be able to: derive mean of sum of random variables; derive expected value of function of two random variables; derive covariance and correlation coefficient; derive variance of sum of random variables; show that the correlation coefficient always lies between -1 and 1; understand the difference between correlation = 0 and independence; understand that correlation is a measure of linear trend. 7.5 Conditional Expectation The conditional expectation (mean) of X given Y = y E ( X | Y = y ) = X x x P X | Y ( x | y ) . . . discrete case = Z -∞ x f X | Y ( x | y ) . . . continuous case In general, E ( g ( X , Y ) | Y = y ) = . . . Notes 20 7.5 Conditional Expectation The conditional mean is the mean of the conditional distribution The conditional variance is the variance of the conditional distribution Notes 20 7.5 Conditional Expectation Example : A business trip is equally likely to take 2, 3, or 4 days. After d -day trip, the change in the traveler’s weight, measure as an integer number of pounds, is uniformly distributed between - d and d pounds. For one such trip, denote the number of days D and the change in weight by W . What are the conditional mean and variance of W | D = 3 ?
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Notes 20 7.5 Conditional Expectation New notation: What is the conditional mean of W given D in our example? i.e., What is E ( W | D ) ? Solve it just like before, but use “D” instead of “3”. Important: The answer will involve D (it will be random!). E [ E ( X | Y )] = Z -∞ Z -∞ x f X | Y ( x | y ) dx f Y ( y ) dy = Z -∞ x Z -∞ f X | Y ( x | y ) f Y ( y ) dydx = Z -∞ x Z -∞ f X , Y ( x , y ) dydx = Z -∞ x f X ( x ) dx = E ( X ) E [ E ( g ( X ) | Y )] = E ( g ( X )) Notes 20 7.5 Conditional Expectation Example : The number N of people who enter an elevator on the ground floor, to go up in Thomas building, is Poisson random variable with mean 4. Thomas building has 5 floors (ignore the basement). Assume that no one enters at any of the other floors. If each person is equally likely to get off at any one of the floors 2 , 3 , 4 , 5 , independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
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