5. Let X have exponential pdf λe_AE 0 when x > 0 otherwise with λ =...
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
6. Write the pdf fy(y; 1) = le-dy, y> 0 in exponential form (see previous problem) and find a sufficient statistic for 1, assuming we have a sample of size n from this pdf.
1 x Suppose X has an exponential distribution, thus its pdf is given by fx (x) = 5e8,0 5x<0, 2> 0;0 0.w. a. Find E(X) b. Find E(X(X-1) c. Find Var (x)
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.
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
3. Let Xi,... , X,n be a random sample from a population with pdf 0, otherwise, where θ > 0. a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ (c) Find the pdf of θ in (b).
A continuous random variable Z is said to have a Laplace(H, b) distribution if its PDF is given by: where μ R and b > 0. a) Ifx-Laplace( 0, b-1), find E(X] and Var[X]. b) If X ~ Laplace(p = 0, b 1) and Y bX + μ, show that Y is a Laplace random variable. c) Let Z ~ Laplace(u, b), where μ E R and b > 0. Find E[2] and Var [2]
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.
Let X be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X > 28]. Answer: CHECK
3 Let (X,Y) be a random vector with the pdf Se-(x+y), f(x,y) = e-(x+y) 122 (x, y) = 1 0, (x,y) E R otherwise. Find P{} <t}. In other words, find the PDF of the r.v. . Done in the class.