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

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- May 5th 2006, 10:53 PMredimoment 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 - May 6th 2006, 12:06 AMrgep
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. - May 6th 2006, 05:53 AMredi
thanks rgep