~ = 0,0² = 1) and let Q(1). = Pr[W1]. (There is no closed-form Let W Normal(μ expression for Q(1), but it's approximately 0.159.) Find the conditional expectation E[W | W> 1] in terms of Q(1).
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- Let X1 ~ x2 (n, 8) and X2 ~ x² (m, 2) : mX1 find the expected value of X = nX2 E (X).Let X be a random variable with pdf f(x) = kx*,-1Let X1,..., Xn be iid pdf 1 f(x) = (log(x)–61)² 262 -00 < x < o0, and unknown parameters 01 and 02. Find the maximum likelihood estimators for 01 and 02, respectively. You do NOT have to do a second derivative test for this exercise.Let X ∼ Uniform (0, 2). Consider a square with sidelength X. (a) Express the area A as function of X. (b) Find the CDF FX(x).(c) Find the PDF fX(x).(d) Determine the expected area of the square with sidelength X? by using the pdf fX(x).Suppose that Y - Normal(0, 1). Compute E[Y}].Let x~ possion(alpha). Show that E[x(x-1)(x-2)....(x-k)]=ak+1Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi . Determine the distribution of ∑i=1 Xi based on its mgf.Let X1 ... Xn i.i.d random variables with Xi ~ U(0,1). Find the pdf of Q = X1, X2, ... ,Xn. Note that first that -log(Xi) follows exponential distribuition.Suppose a value x is chosen "at random" in the interval [0, 1]. In other words, x is an observed value of a standard uniform random variable X ª U(0, 1). Denote by Y the distance of X from the nearest integer. (a) Express Y as a function of X. What is the support of Y? (b) Find P(Y > y). (c) Find both the cdf and the pdf of Y. Can you tell the name of the distribution of Y?SEE MORE QUESTIONS