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April 14th, 2014, 04:43 PM   #1
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Prob derived from mgf

If mgf of x

$\displaystyle m(t)=\frac{e^{5t}-e^{4t}}{t} , t\neq 0$ , $\displaystyle M(0)=1$
a) $\displaystyle E(X)$
seems that $\displaystyle f(x)=1$ since
$\displaystyle \int_4^5 e^{tx}=\frac{e^{5t}-e^{4t}}{t}$
$\displaystyle \int_4^5 xdx=9/2$

b)$\displaystyle Var(X)$
$\displaystyle \int_4^5 x^2dx - 81/4=1/12$

c) find $\displaystyle P(4.2<x\leq 4.7)$

Any errors on a) & b)??

Not sure if I am approaching c) right.

Was thinking

$\displaystyle \int_{4.2}^{4.7}\frac{1}{4.7-4.2}dx$???

I think there is something I am missing on this part.
Thanks for any help!
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April 15th, 2014, 12:00 PM   #2
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c) There is no reason to divide by 4.7 - 4.2. Since f(x) =1 in the interval [4,5], the probability is simply 4.7 - 4.2 = .5 (the integral of f(x) over the interval of interest).
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