Derive the expectation and variance of lognormal distribution WITHOUT using the moment generating functions.
Derive the expectation and variance of lognormal distribution WITHOUT using the moment generating functions.
Derive the moment generating function of the binomial distribution and calculate the mean and variance. P(x) = x = 0,1,2,...,
Derive the moment generating function of the binomial distribution and calculate the mean and variance. p(x)=(*)*(1+p)** x = 0,1,2,...,
2. Consider the Poisson distribution, which has a pdf defined as: a) Derive the moment generating function. b) Use the moment generating function and the method of moments to find the mean and the variance. c) If X follows the Poisson distribution with Xx - 2.3, and Y follows a Poisson distribution with XY-54, what is the distribution of the sum X + Y, assuming that X and Y are independent?
Let X U(0,theta). Find the moment generating function of X and show how to use it to find the mean and variance of X.I think this follows the uniform distribution so..mean = (theta1 + theta2)/2variance = [(theta2- theta1)^2]/ 12moment generating function = [e^(t*theta2) - e^(t*theta1)]/(t * (theta2-theta1))I think the beginnning of the problem means that theta1 is 0? I'm not sure how to show the moment generating function.
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
Let (X,,X2) be jointly distributed with the density function 2-Ax , 0 <x <x, Derive the moment generating function for (X,,X,) 13 a. b. Using the MGF in (a), derive the mean and variance of X, and X, and the correlation coefficient of (x, , x, Let (X,,X2) be jointly distributed with the density function 2-Ax , 0
for university 4- In a lognormal distribution the mean is 3, and the variance is 5. P(x>95)=?
0.4. Suppose that Yi and Y2 are discrete independent random variables with the following moment generating functions: 6 10 102 I. Find the mean and variance of Yi 1- 0.4. Suppose that Yi and Y2 are discrete independent random variables with the following moment generating functions: 6 10 102 I. Find the mean and variance of Yi 1-
7. Derive the moment-generating function M(t) for X 1(a, X). 8. Expand the moment-generating function M(t) = ex+oft®/2 in a power series in t to compute E[X3] if X ~ N(1, 2).
Question 4: Find the variance of the exponential function using the moment generating function. f(x; 1) = {xe-x x>0 10 otherwise