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100AHW7S

# 100AHW7S - STAT 100A HWVII Solution Problem 1 Suppose Z ∼...

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Unformatted text preview: STAT 100A HWVII Solution Problem 1: Suppose Z ∼ N(0 , 1). The density of z is f ( z ) = 1 √ 2 π e- z 2 / 2 . E[ Z ] = 0, Var[ Z ] = 1. Let X = μ + σZ , where σ > 0. (1) Find the probability density function of X . A: Let g ( x ) be the density of X , then g ( x ) = P ( X ∈ ( x,x + Δ x )) Δ x = P ( Z ∈ ( z,z + Δ z ) Δ x = f ( z )Δ z Δ x = f ( x- μ σ ) 1 σ = 1 √ 2 πσ exp {- ( x- μ ) 2 2 σ 2 } . (2) Calculate E[ X ] and Var[ X ]. A: E[ X ] = E[ μ + σZ ] = μ + σ E[ Z ] = μ . Var[ X ] = Var[ μ + σZ ] = σ 2 Var[ Z ] = σ 2 . Problem 2: Suppose U ∼ Uniform(0 , 1). Let T =- log U/λ . (1) For t > 0, calculate P ( T > t ). A: T > t means- log U/λ > t , which is equivalent to U < e- λt . So P ( T > t ) = P ( U < e- λt ) = e- λt . (2) Find the probability density function of T . A: The cumulative density function F ( t ) = P ( T ≤ t ) = 1- P ( T > t ) = 1- e- λt . So f ( t ) = F ( t ) = λe- λt . Problem 3: Consider the following joint probability mass function p ( x,y ) of the discrete random variables ( X,Y ): x \ y 1 2 3 1 .1 .1 .1 2 .2 .1 .2 3 .1 .05 .05 (1) Calculate p X ( x ) for x = 1 , 2 , 3. Calculate p Y ( y ) for y = 1 , 2 , 3. A: p X ( x ) = ∑ 3 y =1 p ( x,y ). p X (1) = . 1 + . 1 + . 1 = . 3. p X (2) = . 2 + . 1 + . 2 = . 5. p X (3) = . 1 + . 05 + . 05 = . 2....
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100AHW7S - STAT 100A HWVII Solution Problem 1 Suppose Z ∼...

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