Find the value of $k$ for given moment generating function.

Question

If $\displaystyle M(t) = k\bigg(\frac{e^t}{7}-5e^t\bigg )^3$, find the value of $k$? What is the variance of random variable $X$?

How do go for to find $k$? I don't know where to begin. Any hint or help will be appreciated.

Answer 1

In such a problem, it is a wise first step to ask 'what properties must be met that will give us a way to solve for an unknown variable such as $k$?'.

If $M(t)$ is a moment generating function then we require that $M(0) = 1$.

So $\displaystyle M(0) = k\bigg( \frac{e^0}{7} -5e^0\bigg)^3 =1$

Or in otherwords, $\displaystyle M(0) = k\bigg(\frac{1}{7}-5\bigg)^3 =1$. This should give you what you need to solve for $k$.

As for variance using an MGF, we have the following : $V(X) = M''(0) - [M'(0)]^2$.

I will leave it to you to complete.