Suppose that X has an exponential distribution with parameter λ. Find the pdf of X2
Suppose X has an exponential distribution with parameter λ = 1 and Y |X = x has a Poisson distribution with parameter x. Generate at least 1000 random samples from the marginal distribution of Y and make a probability histogram.
7. Suppose X1, X2, ..., Xn is a random sample from an exponential distribution with parameter K. (Remember f(x;2) = 2e-Ax is the pdf for the exponential dist”.) a) Find the likelihood function, L(X1, X2, ..., Xn). b) Find the log-likelihood function, b = log L. c) Find dl/d, set the result = 0 and solve for 2.
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
7. (15 pts) Suppose X1, X2, ..., X, is a random sample from an exponential distribution with parameter 2. (Remember f(x;2) = ne-^x is the pdf for the exponential dista.) a) Find the likelihood function, L(X1, X2, Xn). b) Find the log-likelihood function, I = log L. c) Find d //d, set the result = 0 and solve for 2.
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)
You are given the exponential distribution defined for x 〉 0 and λ 〉 0, Mathematically determine the Maximum- likelihood solution for parameter λ if you are given a set of training samples D-XI, X2, , an
2 Let X have an exponential distribution with parameter λ Verify the formulas for expected value and variance on the formula sheet.
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
Let X be an exponential random variable with parameter λ, so fX(x) = λe −λxu(x). Find the probability mass function of the the random variable Y = 1, if X < 1/λ Y = 0, if X >= 1/λ