Question

Help me to solve about moment generating function please? I will use this for study in preparing my midterm of mathematical statistics because I am still low in this topic. Thank you very much for your help before.

Answer 1

`M_X(t)=1/(2-e^t)`.

`E(X)=M'_X(0)=1`.

Explanation:

The moment generating function of a random variable `X` is defined to be the expected value of `e^(tX)`:

`M_X(t)-=E[e^(tX)]`

when `X` is discrete, this becomes

`M_X(t)=sum_("all "x)e^(tx)f(x)`

For `X` as given, we have `x=0,1,2,3,...` so we get

`M_X(t)=sum_(x=0)^oo e^(tx)(1/2)^(x+1)`

Taking a `1/2` out of the sum and combining terms with like powers, we get

`M_X(t)=1/2sum_(x=0)^oo (1/2 e^t)^x`

What's left in the sum is a geometric series which converges to `1/(1-1/2e^t)` as long as `abs(1/2 e^t) < 1`. In other words, as long as `t` is small enough, the geometric sequence can be written in this closed form. (Since MGFs are only concerned with values of `t` near 0, such a `t` can easily be found.)

We now get

`M_X(t)=1/2 [1/(1-1/2 e^t)]`

`color(white)(M_X(t))=1/(2-e^t)`

as our final reduced form of the MGF.

The MGF can be used to find the expected value of `X` (called the first moment) by evaluating its derivative at `t=0`:

`M'_X(t)=d/dt[(2-e^t)^(–1)]`

`color(white)(M'_X(t))=–(2-e^t)^(–2)(–e^t)`

`color(white)(M'_X(t))=e^t/(2-e^t)^2`

So

`E(X) = M'_X(0)=e^0/(2-e^0)^2`

`color(white)(E(X)=M'_X(0))=1/(2-1)^2`

`color(white)(E(X)=M'_X(0))=1 "                     "square`

Verify:

Taking the expected value of `X` in the usual way:

`E(X) = sum_(x=0)^oo xf(x)`

`color(white)(E(X)) = sum_(x=0)^oo x(1/2)^(x+1)`

We would first use something like

`sum_(n=0)^oont^n=sum_(n=0)^oot(nt^(n-1))`

`color(white)(sum_(n=0)^oont^n)=sum_(n=0)^ootd/dt t^n`

`color(white)(sum_(n=0)^oont^n)=td/dt sum_(n=0)^oot^n`

`color(white)(sum_(n=0)^oont^n)=td/dt 1/(1-t)`

`color(white)(sum_(n=0)^oont^n)=t/(1-t)^2`

and then apply it with `t=1/2` and `n=x`:

`E(X) = 1/2sum_(x=0)^oo x(1/2)^x`

`color(white)(E(X)) = 1/2(1/2)/((1-1/2)^2)`

`color(white)(E(X)) = 1/4 [" 1 "/(1/4)]`

`color(white)(E(X)) = 1`