Question

calculate the moment generating function for a random variable which has exponential distribution with parameter gamma.

Answer 1

For an exponential distribution with parameter gamma, therefore the distribution of the random variable here is given as:

X expl

The MGF for X now is obtained here as:

$M_x(t) = \int_{0}^{\infty } e^{xt}f(x ) \ dx$

M.(t) ettye ir da

$M_x(t) = \int_{0}^{\infty } \gamma e^{-x(\gamma - t)} \ dx$

$M_x(t) = \left | \frac{ -\gamma e^{-x(\gamma - t)}}{(\gamma - t)} \right |_{0}^{\infty }$

$M_x(t) =\frac{ \gamma }{(\gamma - t)}$

This is the required MGF for the given random variable here.