Practice Problems for Exam 2 NOTE Solve all questions using only handheld calculator for basic arithmetic operations and the Z table where applies See Lesson 7 notes for the Z table 1 Let X denote the amount of space occupied by an article placed in a 1 ft3 packing container The pdf of X is f x 6x 1 x if 0 x 1 and 0 otherwise Find the expected value and standard deviation of X 2 Arrivals of telephone calls is often modeled as a Poisson process Suppose on average there are 5 calls per hour a If no call has come in the last 30 minutes find the probability that at least one call comes in the next 15 minutes b Suppose it s 10 30 am Find the probability that there will be 10 calls before your break starts at noon 12 00 pm 3 The pdf of X is f x 101 5 1 5 x 4 5e x 10 x 0 Find the variance of X Hint 1 Recognize the distribution of X based on this pdf 4 Let X be a gamma random variable with mean 20 and variance 40 Find P X 15 Hint Determine the parameters and of this gamma distribution 5 The weight of parcels sent in a certain manner is normally distributed with mean 10 lb and standard deviation 1 75 lb Suppose the surcharge is taken if the weight exceeds 12 5 lb What is the probability of surcharge for a randomly selected parcel e x 1 2 20 x Find the 99th percentile of 6 The pdf of X is f x 1 X Hint Recognize the distribution of X based on this pdf 20 7 Let W be uniform on interval 1 2 a Find P W 0 b Find the 6th moment of W c Find E e5W 8 The joint pmf of X and Y is presented in the following table x y 0 1 2 0 0 84 0 06 0 01 1 0 03 0 01 0 005 2 0 02 0 008 0 004 3 0 01 0 002 0 001 a Find the mean and variance of Y b Are X and Y independent Explain c Find the conditional expectation and standard deviation of Y given X 1 d Find the mean of XY e Find E min X Y where min X Y is the smaller of X and Y f Find P 1 XY 2 9 the joint df of X and Y is f x y cx 1 y for 0 y 1 0 x 1 y a Determine c so that f x y is a valid pdf b Find P Y X c Find P Y 1 4 X 1 4 10 Suppose X is a continuous random variable with the pdf f x 1 63x2 for 3 x 6 and Y 1 X 2 Find the pdf of Y