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- X follows a gamma distribution with PDF f(x) = 4xe-2x , where X > 0(a) Derive E(Xn ).If X has an exponential distribution with the param-eter θ, use the distribution function technique to find the probability density of the random variableY = ln X.Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?
- Consider the joint PDF f(x,y)=4xy for 0<x<1 and 0<y<1. (a) Are X and Y independent?(b) Find both marginal distributions. (c) Find both conditional distributions.let x denotes the percentage of time out of 40 hour workweek that a call center agent is serving a client by answering phone calls, suppose that x has probability density function defin by f(x)=3x^2 for 0< x < 1. find the mean and variance of xSuppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?
- What is the Marginal PDF of f(x)=2? Is the event independent?If X is a random variable that is uniformly distributed between -1 to 1, find the PDF √|X| of and pdf of -ln|X|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?
- Find the mean and variance of the gamma distributionusing integration to obtain E(X) and E(X2).[Hint: Express the integrand in terms of a gammadensity.]Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)