# Tut08

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STAT 2011 Probability and Estimation Theory - Semester 1, 2023 Tutorial Sheet Week 8 Tutorial Problems for Week 8 1. Suppose U is a uniform random variable over [0 , 1]. Show that Y = ( b a ) U + a is uniform over [ a, b ] and determine its variance. 2. Let Y be a uniform random variable defined over the interval (0 , 2). Find an expression for the r -th moment of Y about the origin. Also, use the binomial expansion to find E [( Y µ ) 6 ]. 3. Suppose that the random variable Y is described by the pdf f Y ( y ) = cy 4 , y > 1. Find c and determine the highest moment of Y that exists. 4. Let X 1 and X 2 be independent random variables, both having the exponential pdf, f X i ( x i ) = λe λx i , x i > 0, i = 1 , 2. Determine f W ( w ), the pdf of the sum W = X 1 + X 2 . 5. Let f X ( x ) = xe x , x 0, and f Y ( y ) = e y , y 0, where X and Y are independent. Find the pdf of X + Y . 6. Let Y be a continuous non-negative random variable. Show that W = Y 2 has pdf f W ( w ) = 1 2 w f Y ( w ). ( Hint: First find F W ( w ).) 7. If p X,Y ( x, y ) = cxy at the points (1 , 1), (2 , 1), (2 , 2) and (3 , 1), and equals 0 elsewhere, find c . 8. An urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let X denote the number of white chips in the sample and Y the number of blue chips. Write a formula for the joint pdf of X and Y . 9. Let X and Y have the joint pdf f X,Y ( x, y ) = 2 e ( x + y ) , 0 < x < y , 0 < y . Find P ( Y < 3 X ). 10. A man and a woman decide to meet at a certain location. If each of them independently arrives at a time uniformly distributed between 12 noon and 1pm, find the probability that the first to arrive has to wait longer than 10 minutes. 1