MHB Finding the Moment Generating Function

AI Thread Summary
The discussion focuses on finding the moment generating function (MGF) for a given probability distribution. Participants clarify that the correct method involves summing contributions from different intervals and integrating the resulting expression. The limit as t approaches zero should yield M_X(0) = 1, confirming the validity of the MGF. There are some calculations involving L'Hôpital's rule, which leads to confusion about the results. Overall, the importance of correctly summing contributions and applying limits is emphasized for accurate MGF computation.
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I'm working this problem for my math stat class. Here is what I have for it.

First of all, is this the correct method for finding MGF? I thought it was but I don't understand the answers I am getting.

How do I determine my values for t? For both I have t not equal to 0 because t is in the denominator.

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joypav said:
I'm working this problem for my math stat class. Here is what I have for it.

First of all, is this the correct method for finding MGF? I thought it was but I don't understand the answers I am getting.

How do I determine my values for t? For both I have t not equal to 0 because t is in the denominator.

Hey joypav! ;)

More accurately, we have:
$$M_X(t) = E(e^{tX}) = \int_0^2 f(x)e^{tx}dx$$
It seems you have split up the 2 contributions, but we really need their sum.

I haven't checked your calculations, but we will indeed end up with a $t$ in the denominator.
And if we take the limit for $t\to 0$, we should find that $M_X(0)=\lim\limits_{t\to 0} M_X(t) = 1$.
 
I like Serena said:
Hey joypav! ;)

More accurately, we have:
$$M_X(t) = E(e^{tX}) = \int_0^2 f(x)e^{tx}dx$$
It seems you have split up the 2 contributions, but we really need their sum.

I haven't checked your calculations, but we will indeed end up with a $t$ in the denominator.
And if we take the limit for $t\to 0$, we should find that $M_X(0)=\lim\limits_{t\to 0} M_X(t) = 1$.

Okay, that's what I was missing was summing them. Thanks!
 
If we sum them we are just left with 2, correct? x+2-x?

Then when I integrate I get 2(e^2t-1)/t. But when I apply L'hospitals I get 4. I must be making an error.
 
joypav said:
If we sum them we are just left with 2, correct? x+2-x?

Then when I integrate I get 2(e^2t-1)/t. But when I apply L'hospitals I get 4. I must be making an error.

Without checking your calculations, I conclude that:
$$M_X(t) = \int_0^1 e^{tx}xdx + \int_1^2 e^{tx}(2-x)dx
= \left(\frac{e^t}{t} - \frac{e^{t}-1}{t^2}\right) + \left(-\frac{e^t}{t} + \frac{e^{2t}-e^{t}}{t^2}\right)
= \frac{e^{2t}-2e^t + 1}{t^2} = \left(\frac{e^t-1}{t}\right)^2
$$
And:
$$\lim_{t\to 0}\frac{e^t-1}{t} \overset{L'H\hat opital}{=} \lim_{t\to 0}\frac{e^t}{1} = 1
$$
(For the record, mister L'Hôpital was French, and I feel we owe him the respect to at least spell his name properly, although to be fair, when he actually lived, his name was spelled L'Hospital. ;))
 
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