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
The moment generating function of X is the function of t given by the expectation of $\displaystyle \exp(tX)$. Expanded as a power series in t, the coefficient of $\displaystyle t^r$ is $\displaystyle m_r / r!$ where $\displaystyle m_r = {\mathbf E} X^r$ is the r-th moment.
So the expectation of X, which is $\displaystyle m_1$, is the coefficient of t in $\displaystyle M_X(t)$ or alternatively $\displaystyle M'(0)$.
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.