# Math Help - moment generation function

1. ## moment generation function

Need help solving this
The moment generating function of a random variable is given by
M_{X}(t)=(1/6)e^{t}+(1/3)e^{2t}+(1/2)e^{3t.}
a) Find the distribution function of x.
b) Find E(X).
any input highly appreciated.
thx

2. The moment generating function of X is the function of t given by the expectation of $\exp(tX)$. Expanded as a power series in t, the coefficient of $t^r$ is $m_r / r!$ where $m_r = {\mathbf E} X^r$ is the r-th moment.

So the expectation of X, which is $m_1$, is the coefficient of t in $M_X(t)$ or alternatively $M'(0)$.

To recover X we see that the random variable which takes values 1, 2, 3 with probabilities 1/6, 1/3, 1/2 respectively has the required MGF.

There's a good article in MathWorld.

3. thanks rgep