h1 - 1 2 3 4 5 6(a Determine Pr X< 4(b Determine Pr X ≤...

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Stat 5101 (Geyer) Fall 2011 Homework Assignment 1 Due Wednesday, September 14, 2011 Solve each problem. Explain your reasoning. No credit for answers with no explanation. 1-1. For each of the following functions h either determine a constant c such that c · h — that is, the function x 7→ c · h ( x ) — is a PMF or determine that no such constant exists. (a) the identity function on the set { 0 , 1 , 2 } . (b) the identity function on the set {- 2 , - 1 , 0 , 1 , 2 } . (c) the constant function x 7→ 1 on the set { 0 , 1 , 2 } . (d) the constant function x 7→ 1 on the set {- 2 , - 1 , 0 , 1 , 2 } . (e) the function x 7→ x 2 on the set { 0 , 1 , 2 } . (f) the function x 7→ x 2 on the set {- 2 , - 1 , 0 , 1 , 2 } . (g) the function x 7→ x 3 on the set { 0 , 1 , 2 } . (h) the function x 7→ x 3 on the set {- 2 , - 1 , 0 , 1 , 2 } . 1-2. Suppose X is a random variable having the discrete uniform distri- bution on the sample space
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Unformatted text preview: { 1 , 2 , 3 , 4 , 5 , 6 } . (a) Determine Pr( X < 4). (b) Determine Pr( X ≤ 4). (c) Determine Pr(6 < X < 10). 1-3. Suppose X is a random variable having PMF f ( x ) = x 21 , x = 1 , 2 , 3 , 4 , 5 , 6 . (a) Determine E ( X ). (b) Determine E ( X 2 ). (c) Determine E { ( X-3) 2 } . 1 1-4. Suppose X is a Ber( p ) random variable. (a) Show that E ( X k ) = p for all positive integers k . (b) Determine E { ( X-p ) 2 } . (c) Determine E { ( X-p ) 3 } . 1-5. 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 } . 2...
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h1 - 1 2 3 4 5 6(a Determine Pr X< 4(b Determine Pr X ≤...

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