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September 10th, 2017, 06:30 PM   #1
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Exclamation Moment generating function for a random variable described by a piecewise pdf



Part e) is the problem in question here; I'm not exactly sure how to find the moment generating function, specifically when 1 <= y < 2. I know that M(t) = the integral from -infinity to infinity of e^(tx)*f(x)*dx, but do I simply use 2-y for f(x) on that interval? I'm not exactly sure, I think that I have to account for the previous interval in some way first. Would somebody please offer some direction? Thank you.
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September 10th, 2017, 07:39 PM   #2
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$M_Y(t) = E[e^{t Y}] = \displaystyle \int~f_Y(y)e^{t y}~dy$

$M_Y(t) = \displaystyle \int_0^1~y e^{t y}~dy + \int_1^2~(2-y)e^{ty}~dy = \dfrac{\left(e^t-1\right)^2}{t^2}$

Now

$E[Y^n] = M_Y^{(n)}(0)$

This may have to be evaluated as a limit.

in particular

$E[Y] = \displaystyle \lim_{t \to 0} \dfrac{d}{dt}M_Y(t)(0) =\lim_{t \to 0} \dfrac{2 \left(e^t-1\right) \left(e^t (t-1)+1\right)}{t^3} $

I leave it to you to calculate that.

Finally

$V(Y) = E[Y^2]-\left(E[Y]\right)^2$

and $E[Y^2] = \displaystyle \lim_{t\to 0} \dfrac{d^2}{{dt}^2} M_Y(t)$

You should have all the pieces you need now.
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September 11th, 2017, 05:58 PM   #3
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I really appreciate the help. It was the second step in your solution that I was struggling to obtain. The rest of it is straightforward from that point, but it's pretty brutal.
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September 11th, 2017, 06:26 PM   #4
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Quote:
Originally Posted by John Travolski View Post
I really appreciate the help. It was the second step in your solution that I was struggling to obtain. The rest of it is straightforward from that point, but it's pretty brutal.
I'm curious if you are allowed to use software to do the integrals and derivatives.

While it's important to learn to work the integrals of Calc 2 by hand, once that's done, in this day and age I'd think it just fine to use software to compute them.

You aren't learning anything about the moment generating function by grinding through these integrals and derivatives here.
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September 11th, 2017, 07:32 PM   #5
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I honestly don't know (I doubt it since we can't even use calculators on the exams), I'll have to ask my instructor tomorrow.

You should see some of our other problems... he showed us an exam question from last semester and the solution was to differentiate an infinite series multiple times to find the value that the sum converged to (in order to find V(x) for a discrete random variable). The calculus alone took up the entire page.

Considering that I have no idea how many problems are going to be on an exam, I'm really concerned about this class.

Last edited by John Travolski; September 11th, 2017 at 08:19 PM.
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