Probability and Moment Generating Functions

**arguabsysi** · May 22nd 2008, 02:16 AM

Hello,

I am terrible when it comes to probability and moment generating functions. I missed 2 lectures this week and I am now completely lost on the topic.I have two questions that I need to do by tomorrow and I was hoping if shown how to do one of the questions, I could work out the other one myself as it is similar.

Here is the first question:

Let Xn be a discrete random variable that takes the values 1,2,...,n with equal probability 1/n. Find the probability generating function of Xn and then determine its moment generating function. Determine the moment generating function of Yn = Xn/n. Show that the moment generating function of Yn = Xn/n converges pointwise to the moment generating function of a random variable that is uniformly distributed on (0,1).

I know this is a lot to do, so even if someone could tell me what it is I have to do here would be great. Like I said, I missed 2 lectures so Ive tried to learn this concept on my own, and have had nobody to correct any assumptions I make that are incorrect. Thank-you.

**mr fantastic** · May 22nd 2008, 02:43 AM

Originally Posted by arguabsysi

Hello,

I am terrible when it comes to probability and moment generating functions. I missed 2 lectures this week and I am now completely lost on the topic.I have two questions that I need to do by tomorrow and I was hoping if shown how to do one of the questions, I could work out the other one myself as it is similar.

Here is the first question:

Let Xn be a discrete random variable that takes the values 1,2,...,n with equal probability 1/n. Find the probability generating function of Xn and then determine its moment generating function. Determine the moment generating function of Yn = Xn/n. Show that the moment generating function of Yn = Xn/n converges pointwise to the moment generating function of a random variable that is uniformly distributed on (0,1).

I know this is a lot to do, so even if someone could tell me what it is I have to do here would be great. Like I said, I missed 2 lectures so Ive tried to learn this concept on my own, and have had nobody to correct any assumptions I make that are incorrect. Thank-you.

This will get you started:

Probability generating function: Read Generating Functions.

So $G_{X_n} (t) = E(t^{X_n}) = \sum_{j = 1}^{n} \frac{t^j}{n} = \frac{1}{n} \sum_{j = 1}^{n} t^j$ .

Moment generating function: Read Moment-generating function - Wikipedia, the free encyclopedia.

So $m_{X_{n}} (t) = E \left( e^{tX} \right) = \sum_{j = 1}^{n} \frac{e^{jt}}{n} = \frac{1}{n} \sum_{j = 1}^{n} e^{jt}$ .

The moment generating function of Y = aX + b is $m_Y (t) = e^{bt} \, m_X(at)$ : See Moment Generating Function.

In your question $a = \frac{1}{n}$ and b = 0.

The moment generating function of a random variable distributed uniformly on (a, b) is $\frac{e^{bt} - e^{at}}{t(b - a)}$ : See Uniform distribution (continuous) - Wikipedia, the free encyclopedia.

Substitute a = 0 and b = 1 to get the answer your shooting for when finding $\lim_{n \rightarrow \infty} m_Y (t)$ ......

**arguabsysi** · May 22nd 2008, 03:11 AM

Thank-you so much. I'll see what I can come up with.

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