If X is , it is known that Z = is χ2(1). a) Prove that E(Z) = 1. b) The PDF of a random variable which has a Gamma(α,ϑ)

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If X is , it is known that Z = is χ2(1).
a) Prove that E(Z) = 1.
b) The PDF of a random variable which has a
Gamma(α,ϑ)(α,ϑ) is
f(x)=1r(α)θαxα−1e−xθ0≤x<∞,α>0,θ>0f(x)=1r(α)θαxα−1e−xθ0≤x<∞,α>0,θ>0

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6. If X is, it is known that Z = is x2(1). a) Prove that E(Z) = 1. b) The PDF of a random variable which has a Gamma(a, 0) is f(x) = -xa-1 e Plage a 0 < x < 0, a > 0, 0> 0 x?(1) is equivalent to a random variable that is Gamma(1/2, 2). Find the mean and variance of a random variable that is x2(1). c) The sum of k independent x2(1) random variables has a x?(n-1) distribution. The expression: (n-1)s2 can be shown to be the sum of n-1 independent x2(1) random variables. Find the mean and variance of the estimator, o2. 02