5. (11 pts) Use the Distribution Function Method on this
problem: The random variable Y has an exponential distribution with
parameter β. Let ?? = √??. Find the pdf of U. Note: U has a Weibull
distribution. You will see the Weibull distribution many times in
this course
The random variable Y has an exponential distribution with parameter β.
Find the pdf of U.
U =
5. (11 pts) Use the Distribution Function Method on this problem: The random variable Y has...
(11 pts) Use the Distribution Function Method on this problem: The random variable Y has an exponential distribution with parameter B. Let U vY. Find the pdf of U. Note: Uhas a Weibull distribution. You will see the Weibull distribution many times in this course. 5.
6. (11 pts) Use the Distribution Function Method here: The
random variable ??~????????(∝= 4, ?? = 2). Let ?? = ??4. Find the
pdf of U.
(1 l pts) Use the Distribution Function Method here: The random variable Y~Beta(α= 4, β = 2). Let U 6. Y4, Find the pdf of U.
(11 pts) Use the Distribution Function Method here: The random variable y-Beta( Let U Y4. Find the pdf of U. 4,β-2). 6.
6. (11 pts) Use the Distribution Function Method here: The random variable Y~ Beta(o4,B 2). Let U-Y4. Find the pdf of U.
2- 5. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. Let B>0, 8>0. Consider the probability density function x>0 zero otherwise Recall (Homework #1) V-Χδ has an Exponential(8-T )-Gamma(u-l,e-1 ) distribution. Let X1, . , X/ be a random sample from the above probability distribution. y-ΣΧ.Σν i has a Gamma(u-n, θ- 1 ) distribution. !!! i-l 2. suppose δ is known. Let Xi, X2, , Xn be a random sample from the distribution with...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF
Problem 5. Let X...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
7. (11 pts) Use the Transformation Method on this problem (be
sure to verify that the function h(y) is increasing or decreasing
over the domain of y, either by graphing h(y) or by using
differential calculus):
7. (11 pts) Use the Transformation Method on this problem (be sure to verify that the function h(yjs increasing or decreasing over the domain of y, either by graphing h(y) or by using differential calculus): The random variable Y~Gamma(o:: 3/2,β-4). Use the transformation method...
4. (9 pts) Suppose the random variable Y has a geometric
distribution with parameter p. Let ?? = √?? 3 3 . Find the
probability distribution of V
3 4. (9 pts) Suppose the random variable Y has a geometric distribution with parameter p. Let V 3 Find the probability distribution of.
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)