х, Osx<1 S(x)={2-x, 1Sx<2 0, else (1) Find the distribution function F(x). e) Caleulate Pex< (3) Calculate E[X] and Var(X).
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Let X be a continuous random variable, with probability density
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- If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.a) Find the marginal pmfs of X and Y b) Find the conditional pmf of X given Y = 1
- A system consists of two components whose lifetimes (in years) X1, X2 are inde-pendent and identically distributed with pdf f(x) = 0.5e−0.5x, x >0. (a) Show that the mean of X is 2 and use this to find E(X2−2X1). (b) ComputeP(X2= 2X1) andP(X2>2X1). (c) ComputeCov(X1, X2−2X1). You may use the fact that Var(X1) = 4.Take the model y = y x'1β1+x'2β2+ϵ with E(xe)=0 Suppose B1 is estimated by regressing Y on X1 only. Find the probability limit of this estimator.For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)variance
- If X has the exponential distribution given by: f(x) = 4*(1 - x/2); 1 ≤ x ≤ 2. Find E(x).LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;2 Given the exponential distribution: f(x) = {1/θ e^(-x/θ) ; x > 0 0; ew]
- X follows a gamma distribution with PDF f(x) = 4xe-2x , where X > 0(a) Derive E(Xn ).If X is an exponential random variables with rate 1, then its distribution function is given by F(x) = 1 − e−x Show that x = − ln(1 − u).f(x) for a continuous probability function is 1/18 , and the function is restricted to 2 ≤ x ≤ 20. What is P(x < 2)?