[Math] Finding moment generating function from piecewise CDF

actuarial-scienceprobabilitystatistics

I am stuck on this problem:

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I am given the CDF, and unless there is a shortcut I am not remembering, I need to find the PDF before I can get the moment generating function and solve the problem.

In this case, it looks like the only part that "matters" is the second part, because if f(x) = (d/dx)F(x), the x<1 and x>=2 pieces are just zero. Thus, I think I am left with this PDF: f(x) = (x-1) for 1 <= x < 2. To get the moment generating function, then, I would do this, right? :

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If I calculated right, this integral gives me:

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I think the second derivative of this would have t stuff in the bottom, so how could I evaluate it at zero (if that is what is being asked for)? Otherwise, where did I go wrong in approaching this?

Best Answer

The second derivative of the MGF evaluated at 0 is the second moment of the RV, hence the name. So your integral should become easier.

So we have: $E[X^2] = \frac{1}{2} + \int_1^2 t^2 \cdot (t-1) dt = \ldots = \frac{23}{12} \approx 1.92$

Note that the CDF has point mass $\frac{1}{2}$ at $1$, hence the $\frac{1}{2}$.

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