For the density function f(x) = λe^(-λx) for x>=0, find EX
Random variable X has the pdf f(x) = λe^(−λx) for x > 0. (a) Derive the CDF of X. (b) Derive the moment generating function of X. (c) Derive the mean of X. (d) Derive the variance of X.
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).
Recall that X ∼ Exp(λ) if the probability density function of X is fX(x) = λe−λx for x ≥ 0. Let X1, . . . , Xn ∼ Exp(λ), where λ is an unknown parameter. Exponential random variables are often used to model the time between rare events, in which case λ is interpreted as the average number of events occurring per unit of time. Recall that X ~ Exp(A) if the probability density function of X is fx(x)-Ae-Az for...
In question 5, f(x) = λ*exp(-λx), for x greater or equal to 0, and zero otherwise. 9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
5. (20%) Let X be a continuous random variable whose probability density function is fr(x) (a +bx)%0(x) (a) If Ex)f find a and b. (b) Give the cumulative distribution function F,(x) f()dt of X and Var(X) (c) Let A be any Borel set of R. Define P by P(A) [,f dm 5. (20%) Let X be a continuous random variable whose probability density function is fr(x) (a +bx)%0(x) (a) If Ex)f find a and b. (b) Give the cumulative distribution...
. A random variable X with P(X> 0) 1 has density function f(x) cx299e3. Find: with P(X >0) 1 has density function f (x)cx2s e ) E(X) ) Var(x)
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
density function f(x; θ)-829-1, 0 < x < 1, 0 < θ < oo. Find the MLE θ
1. A random variable X has the cumulative distribution function exe F(X) = 1 + ex • Find the probability density function • Find P(0 < X < 1)