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# 100AHW5 - X prove Var X = E X 2-E X 2 Please prove it for...

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STAT 100A HWV Due next Wed in class Problem 1: For a discrete random variable X , prove (1) E[ aX + b ] = a E[ X ] + b . (2) Var[ aX + b ] = a 2 Var[ X ]. Problem 2: For two discrete random variables X and Y , if X and Y are independent, prove (1) E[ X + Y ] = E[ X ] + E[ Y ]. (2) Var[ X + Y ] = Var[ X ] + Var[ Y ]. Problem 3: For Z Bernoulli( p ), prove E[ Z ] = p and Var[ Z ] = p (1 - p ). Problem 4: For X Binomial( p ), prove (1) E[ X ] = np , Var[ X ] = np (1 - p ). (2) E[ X/n ] = p , Var[ X/n ] = p (1 - p ) /n . (3) Argue that X/n p as n → ∞ . Problem 5: For a random variable
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Unformatted text preview: X , prove Var[ X ] = E[ X 2 ]-E[ X ] 2 . Please prove it for both discrete and continuous cases. Problem 6: For U ∼ Uniform[0 , 1], (1) Calculate cumulative density function F ( u ) = P ( U ≤ u ). (2) Calculate E[ U ], E[ U 2 ], and Var[ U ]. Problem 7: For T ∼ Exponential( λ ), (1) Calculate F ( t ) = P ( T ≤ t ). (2) Calculate E[ T ]. 1...
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