Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + B) r(a)r(3)* r^-1(1−z)ß-1, 0 < x < 1, ~ Beta(a +3,7), namely its PDF I(a +8+y) fy (y) = = -yª+³−¹(1 — y)²-¹, - 0 < y < 1, r(a + 3)(y) nd X and Y are independent. Define U = XY, V = X. Find the joint PDF fuv(u, v).

Let X₁,..., Xn be an iid random sample from a Uniform(0, 0), where 0 > 0. Let X(n) = max{X₁, (a) Find the mean and variance of X(n). (b) Find the limiting distribution of Zn = n (0 – X (n)). (c) Find the limiting distribution of log(Zn). Xn}

Let X (X1, X2,..., Xn) be a sample of i.i.d. Poisson(0) random variables with probability mass %3D function e-gz f(x, 8) = Po(X = x) = æ! x € {0,1, 2, .}, 0 > 0. The statistic T(X) = Ei-1Xi is complete and sufficient for 0. i3D1 Derive the UMVUE of h(0) = e_kU where k = 1, 2, ... , n is a known integer. Hint: Use the interpretation that P(X1= 0) = e-º and therefore we have that P(X1 = 0,.., Xk = 0) = P(X1 = 0)* = e¬k® and also note that n T(X) = X; ~ Poisson(n0). i=1 (-) Pumvue 1 - 1 ( t Îumvue Îumvue = (-)' humvue (-)" Îrumvue

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