Let X be a Unif(−6,6) variable, that is, X is Uniformly distributed over the interval (−6,6).
Let $X_1,X_2,…,X_n$ be independent Unif(−6,6) variables. Let
$Y=\sum_{i=1}^nX_i$
Find the mgf $M_Y(t)$ of Y. Evaluate the mgf at the point t=0.28 in the case n=5
so for this i know using the property of MGF it would be just multiples of the 5 IID MGFs
Standardize Y to create a new variable Z with mean zero and standard deviation 1. Find the mgf $M_Z(t)$ of Z. Evaluate $M_Z(t)$ at the point t=2.95 in the case n=11.
for this question how am I going to approach to this? could anyone give me some hints? It would be grateful if someone could guide me through this.
Best Answer
for the first question, we go back to basics. For any one of the $X_i$, we have that the mgf of $X_i$ is equal to $$E(e^{tX_i}).$$ The density function of $X_i$ is $\frac{1}{12}$ on $(-6,6)$ and $0$ elsewhere. Thus $$E(\exp(tX_i))=\int_{-6}^6 \frac{1}{12}e^{tx}\,dx.$$ Integrating, we find that the mgf is $$\frac{1}{12t}\left(e^{6t}-e^{-6t}\right).$$ Take the fifth power of this to find the mgf of $Y$.
For the second problem, we first need to find the mean and variance of $Y$. The mean is $0$ (good, there is nothing we need to do). For the variance of $Y$, this is equal to $5$ times the variance of any of the $X_i$.
The variance of the uniform on $(-6,6)$ is $\frac{1}{12}(6-(-6))^2$, that is, $12$. It follows that $Y$ has variance $60$.
Thus $Z=\dfrac{Y}{\sqrt{60}}$, since multiplying a random variable by $k$ multiplies the variance by $k^2$.
We found the moment generating function $M_Y(t)$ of $Y$. The moment generating function of $Z$ is given by $$M_Z(t)=M_Y(t/\sqrt{60}).$$