6. Write the pdf fy(y; 1) = le-dy, y> 0 in exponential form (see previous problem)...
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0 otherwise Let U1 = min{Y1, ... ,Yn} and U2 = $1=1 Yį. (a) Show that the pdf of Uı is f(y) = ne-n(y=0)1(y > 0) (b) Show that U1 - 1/n and U2/n - 1 are both unbiased estimators of 0. (c) Find the variance of each of the unbaised estimators in part (b). (d) One of U1,U2 is a sufficient statistic. Which one?...
5. Let X have exponential pdf λe_AE 0 when x > 0 otherwise with λ = 3. Let Y-LX). Find E(Y) and Var(Y)
Exponential(). That is Y has a density function of the form 7. Let Y Ay f(y) = de"^9,y> 0 where 0. Show that: (a) If a >0 and b > 0, then P(Y > a + b|Y > a) = P(Y > b) (b) E(Y) 1/A
Suppose X and Y are independent random variables with Exponential(2) distribution (Section 6.3). We say X ~ Exponential(2) if its pdf is f(x) = -1/2 for x > 0.
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).
The random Variable X has a pdf fx (2) = {*** kr + > -1 <r<2 otherwise Y is a function of X and is derived using Y = g(x) = X S -X X2 X <0 X>0 Find: (A) fr(y) (B) E[Y] using fy(y) (C) EY] using fx (2)
Let Y be a random variable with probability density function, pdf, f(y) = 2e-2y, y > 0. Determine f (U), the pdf of U = VY.
Example 7. Let Y1, ... ,Yn be a random sample from a Rayleigh distribution with pdf Ske-?/(20) f(y\C) = 10 = if y>0,0 > 0 otherwise otherwise Find a sufficient statistic for 0.
Problem 7: Let X and Y be two jointly continuous random variables with joint PDF 4 (x y) otherwise a) Find P(0< Y< 1/2 I x-2) b) For what value of A is it true that P(0 < Y < ½ |X> A)-5/16
Use logarithmic differentiation to find dy/dx. y = XV x2 + 25 X>0 dy - dx Need Help? Read It Talk to a Tutor