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If a random variable has a moment generation function $$M(t) = a + (1-a) \, e^t$$ with $ 0 < a < 1$

a) Determine the Distribution of $X$

b) Show that the moment of r-class of $X$ is equal to $E[X^r] = 1-a$ , for $1,2,\ldots$

Edit:

So this is my best best approach till now; We know that $$M_x(t) = \mathbb{E} \left[e^{tx} \right] = \int_{ -\infty}^{+\infty} e^{t x} f(x) dx $$

Also I know that $M_x(t=0)=1$.

So, if I could find $f(x)$, I could easily then find $F(x)$ (distribution).

The problem is that I am doing cyrcles all over my papers .

Should I find $\text{Var}(X)$? thats easy... and then try from this?

Can't think of anything else.

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    This is not a "do my homework for me" site. Show us what you've tried and where are you stuck. And dont' copy the exercise statement, as if given orders. Read http://math.stackexchange.com/questions/how-to-ask http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question2012-06-02
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    @ leonbloy never said that this is a "do my homework" site ...2012-06-03
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    Back on the problem , it seems that the Var(X) equals to zero ... also, all the derivatives of Mx(t) are all equal .2012-06-03
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    I edited the math, check if all is ok.2012-06-03
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    "Var(X) equals to zero " Why?2012-06-03
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    yeah right wrong calculation2012-06-03

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