X = Beta(
)
Y = 1- X follow Beta()
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X....
Suppose X ~ Beta(a, β) with the constants α,β > 0, Define Y- 1- X. Find the pdf of Y.
7. Suppose X and Y have joint pdf f(x,y) = 24x y if x >0, y>0,x+y<1 and 0 otherwise. Find P(Y > 2x).
b. Suppose ~ Γ(α, β), with α > 0, β > 0 and let Y-eu. Find the probability density function of Y Find EY and var(Y)
Q3. Find the quantile function F-1 for F(r)-1-1-α, x > 1.
2. Suppose an exact linear relationship exists between two random variables X and Y That is, let Y-α + ßx, where α and β are constants and β > 0. Prove that ρχ,-1 Hint: Substitute α + βΧ into the formula for Pry and apply the covariance rules.
. For > 0 and A > 0, define the joint pdf -Ay = 0<x<A,<y, fx.y(,y) 10 else. (a) Express c in terms of X and A. (b) Find E[XY]. (c) Let [2] be the largest integer less than or equal to z. For example, (3.2] = 3 and [2] = 2. Find the probability that [Y] is even, given that 4 <x< 34
5.4.3 Suppose that X,, X" are iid exponential with mean β(> 0), , y, are iid exponential with mean η(> 0), and that the A's are independent of the Ys. Define ½,-. Then. m, n (i) Show that I , is distributed as /m (ii) Determine the asymptotic distribution of T m, n 2m, 2n as n ->o, when is m, n kept fixed; (ii) Determine the asymptotic distribution of T, as m, when n is kept fixed.
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
3. Suppose that X has pdf fx(x) = 3, x > 1 and Y has pdf 24» fy(y) = ¡2, x 〉 1. Suppose further that X and Y are inde- pendent. Calculate the P(X 〈 Y).
For f(x, y) = k(x2 + y2), 0<x< 1 and 0 <y<1 and 0 elsewhere: a) Find k. b) Are X and Y independent? c) Find P(X<0.5, Y>0.5), P( X = 0.5, Y>0.5).