h2 - Statistics 5101 Fall 2010 Homework Assignment 2 1...

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Statistics 5101, Fall 2010: Homework Assignment 2 1. Suppose X is a random variable having the discrete uniform distribution on the sample space { 1 , 2 , 3 , 4 , 5 , 6 } . (a) Determine Pr( X < 4). (b) Determine Pr( X 4). (c) Determine Pr(6 < X < 10). 2. Suppose X is a random variable having PMF f ( x ) = x 21 , x = 1 , 2 , 3 , 4 , 5 , 6 . (d) Determine E ( X ). (e) Determine E ( X 2 ). (f) Determine E { ( X - 3) 2 } . 3. Suppose X is a Ber( p ) random variable. (g) Show that E ( X k ) = p for all positive integers k . (h) Determine E { ( X - p ) 2 } . (i) Determine E { ( X - p ) 3 } . 4. Determine the set of real numbers θ such that f θ ( x ) = θ, x = x 1 θ 2 , x = x 2 1 - θ - θ 2 , x = x 3 is a PMF on the sample space { x 1 , x 2 , x 3 } . 5. Suppose we have a PMF f with domain S (the original sample space), and we have a map g : S Ω that induces a probability model with PMF pr with domain Ω (the new sample space) given by the formula on slide 107. Prove that for any real-valued function h on Ω X ω Ω h ( ω ) pr( ω ) = X x S h ( g ( x ) ) f ( x ) 6. Suppose pr is the uniform distribution on Ω = {- 2 , - 1 , 0 , 1 , 2 } and the random variable X
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