Question

Let X be a continuous random variable with probability density function: $f_X(x) = \begin{cases} xe^{-x}, & x \ge 0 \\ 0, & \text{else} \end{cases}$ (a) Show that the moment generating function of X is $M_X(t) = \frac{1}{(1-t)^2}$. (b) Compute E[X] using $M_X(t)$. (c) Compute Var(X) using $M_X(t)$.

          Let X be a continuous random variable with probability density function:
$f_X(x) = \begin{cases} xe^{-x}, & x \ge 0 \\ 0, & \text{else} \end{cases}$
(a) Show that the moment generating function of X is $M_X(t) = \frac{1}{(1-t)^2}$.
(b) Compute E[X] using $M_X(t)$.
(c) Compute Var(X) using $M_X(t)$.

Let X be a continuous random variable with probability density function:
fX(x) = xe^-x, x ≥ 0
0, else
(a) Show that the moment generating function of X is MX(t) = (1)/((1-t)^2).
(b) Compute E[X] using MX(t).
(c) Compute Var(X) using MX(t).

Added by Jeffrey H.

Mathematical Statistics and Data Analysis

John A. Rice 3rd Edition

Chapter 4

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Let X be a continuous random variable with probability density function : ce- x 0 else (a) Show that the moment generating function of X is Mx (t) = (b) Compute E[X using Mx(t) (c) Compute Var(X) using Mx(t).