Let a pdf of random vector X be given as fx(x) = K exp(-xa)u(x) where a = [a₁, ...‚ an]. Find the value of K.
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- Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.Let a random variable X has the CDF as;Fx(x) = 1 ; x ≥ 1 1/2+x ; 0 ≤ x < 1 0 ; x < 0 Verify that Fx(x) is a CDF. Also- (i) find the PDF of X (ii) Calculate E[ex ] (iii) Determine the probability P[X = 0|X ≤ 0.5]Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.
- Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.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;Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
- Let X be a continuous random variable with a pdf f(x) = { kx5 0≤x≤1, 0 elsewhere Determine the value of k.let X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
- Let X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdf f(x)={1/9(4-x^2) -1<=x<=2 0 otherwise a) Compute P(0≤ X ≤1)b) Obtain E(x) and Variance of XLet X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?