- Thread starter
- Moderator
- #1

- Jan 26, 2012

- 995

Thanks again to those that participated in the second round of our POTW! Now, it's time for the third one!

This week's problem was proposed by yours truly.

-----

\[f(x) = \left\{\begin{array}{cl}\lambda e^{-\lambda x} & x\geq 0,\,\lambda >0\\ 0 & x<0\end{array}\right.\]

Show that $\sum_{i=1}^n X_i$ is equivalent to a random variable of the Gamma distribution $\Gamma(n,\theta)$, where the p.d.f. of the Gamma distribution is given by

\[f(x) = \left\{\begin{array}{cl}\frac{1}{\theta^n\Gamma(n)}x^{n-1}e^{-x/\theta} & x\geq 0,\,\theta>0,\, n\in\mathbb{Z}^+\\ 0 & x<0\end{array}\right.\]

-----

Here are two hints:

If $X$ is a continuous random variable, we define the

\[M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty}e^{tx}f(x)\,dx\]

where $f(x)$ is the p.d.f. of the random variable $X$. Use the fact that if $\{X_i\}_{i=1}^n$ is a collection of random variables, then

\[M_{\sum_{i=1}^n X_i}(t) = \prod_{i=1}^n M_{X_i}(t)\]

Remember to read the POTW submission guidlines to find out how to submit your answers!

This week's problem was proposed by yours truly.

-----

**Problem**: Let $X_i,\, (i=1,\ldots,n)$ be a (continuous) random variable of the exponential distribution $\text{Exp}(\lambda)$, where it's probability density function (p.d.f.) is defined by\[f(x) = \left\{\begin{array}{cl}\lambda e^{-\lambda x} & x\geq 0,\,\lambda >0\\ 0 & x<0\end{array}\right.\]

Show that $\sum_{i=1}^n X_i$ is equivalent to a random variable of the Gamma distribution $\Gamma(n,\theta)$, where the p.d.f. of the Gamma distribution is given by

\[f(x) = \left\{\begin{array}{cl}\frac{1}{\theta^n\Gamma(n)}x^{n-1}e^{-x/\theta} & x\geq 0,\,\theta>0,\, n\in\mathbb{Z}^+\\ 0 & x<0\end{array}\right.\]

-----

Here are two hints:

**moment generating function**by

\[M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty}e^{tx}f(x)\,dx\]

where $f(x)$ is the p.d.f. of the random variable $X$. Use the fact that if $\{X_i\}_{i=1}^n$ is a collection of random variables, then

\[M_{\sum_{i=1}^n X_i}(t) = \prod_{i=1}^n M_{X_i}(t)\]

Recall that $\Gamma(x) = \displaystyle\int_0^{\infty}e^{-t}t^{x-1}\,dx$.

Remember to read the POTW submission guidlines to find out how to submit your answers!

**EDIT**: I forgot to mention that each $X_i$ are i.i.d. random variables. If they're not, then the above result doesn't hold (thanks to girdav for pointing this out).
Last edited: