Let X and Y be independent and normally distributed RVs with the same PDF Find the PDFS of: (i) X+ Y; (ii) x: X2: (iii) x²+ y?.
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- Let X1, X2 denote two independent variables, each with a x^2(2) distribution. Find the joint pdf of Y1=X1 and Y2 = X2+X1. Note that the support of Y1, Y2 is 0<y1<y2<infinity. Also, find the marginal pdf of wach Y1 and Y2. Are Y1 and Y2 independent?Suppose that {Xk} k=1,...,n are i.i.d ∼ Exp(λ), the pdf is given by f(x) = λe^(-λx), x > 0. Let Y1 be the 1st–order statistics. Given a > 0, evaluate P(Y1>a/n)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?
- Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Find the pdf of e^x in terms of the pdf of X, base the answer to the case where X is uniformly distributed between 0 and 1Given that X1, X2, and X3 are standard normal IID RVs. Find the joint pdf of (Y1, Y2) which are defined below: Y1 = X1 + X2 + X3 and Y2 = X1 + 2Z2 - 3Z3
- Let X have cdf Fx(x) = 1- (1+x)^-A, x,a > 0. Determine the pdf and cdf of Y = CXLet X be a continuous random variable with a pdf f(x) = { kx5 0≤x≤1, 0 elsewhere Determine the value of k.Suppose that a person’s score X on a mathematics aptitude test is a number between 0 and 1 and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in Belize, the scores X and Y redistributed according to the following joint pdf. f(x,y) = 2/5(2x + 3y) for 0≤x≤1, 0≤y≤1 and 0 otherwise What proportion of college students obtain a score greater than 0.8 on the mathematics test? If a student’s score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?