Distribution MGF ψ(t)Bernoulli(p)pet+(1−p)Binomial(n,p)(pet+(1−p))nPoisson(λ)eλ(et−1)Normal(μ,σ)exp{μt+σ2t22}Gamma(α,β)(11−βt)α for t<1/β MGF 为什么能用? 已知: \begin{equation} e^{t x}=\dfrac{(t x)^{0}}{0 !}+\dfrac...
Let X_1, ..., X_n be a random sample from the Negative Binomial distribution f(x, p) = \binom{x - 1}{3 - 1}p^3 (1 - p)^{x - 3}, x = 3, 4, ... a. Derive the MLE \hat{p} of p, based on the distribution Let...
Let X have the γ distribution with parameters α and θ. 1. Find the moment-generating function of X. 2. Find the mean and variance using the MGF of X. Gamma Distribution: The gamma distribution is the general version of ...
Sometimes, it is easier to work with the moment generating function than with actual density functions or cumulative distribution functions. Answer and Explanation: The mgf of the random variable Y is given by: $$M_Y(t) = E(e^{tY})=E(e^{tX^2...
Suppose that X sim Binomial(n,p) and Y sim Binomial(m,p) and that X and Y are independent. (a) Find the moment-generating function (mgf) of X . (b) Determine the distribution of Z = n - Suppose that X sim N (mu...
Assume that Xi \sim Binomial(ni , p) for i = 1, 2 and X1 and X2 are independent. Find the conditional distribution of X1|(X1 + X2). Let X \sim U(-1, 1) and Y \sim Exp(2), and suppose that X and Y a...