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Unformatted text preview: Practice problems for Midterm 2 Problem 1 The amount X in pounds of polyurethane cushioning found in a car is modeled as a continuous random variable with pdf f X ( x ) = 1 ln2 1 x if 25 x 50 otherwise . ( a ) Find the mean, variance and standard deviation for X . ( b ) Let Y = 1 /X . Find the mean of Y . Problem 2 The joint density of a continuous random vector ( X,Y ) is given by f X,Y ( x,y ) = 1 10 (4 x + 2 y + 1) if 0 < x < 1 and 0 < y < 2 otherwise . ( a ) Compute Cov( X,Y ) and the correlation X,Y . ( b ) Find the number a that minimizes the variance of the linear combination aX + Y . Problem 3 A vaccine for desensitizing patients to bee stings is to be packed with three vials in each box. Each vial is checked for strength before packing. The probability that a vial meets specifications is . 9. ( a ) Let X denote the number of vials that must be checked to fill a box. Find the pmf of X , its mean and its variance....
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