Mathematics, 19.12.2019 19:31 crispingolfer1864
Let x unif[0; 1]. parts (a) and (b) below ask you to nd the nth moment of x in two di erent ways. (a) use the formula e(xn) = r 1 0 xn dx. (b) compute the moment generating function mx(t) (or use the result of exercise 5.3). find the taylor series expansion of mx(t) and identify the coecients. hint. mx(t) will be written in terms of et. substitute the taylor series for et into the formula and simplify the expression.
Answers: 3
follows the standard uniform distribution, so 
is 1 if ![x\in[0,1]](/tpl/images/0426/3202/85477.png)
and 0 otherwise. Then the 
th moment of ![E[X^n]=\displaystyle\int_{-\infty}^\infty x^nP(X=x)\,\mathrm dx=\int_0^1x^n\,\mathrm dx=\dfrac{x^{n+1}}{n+1}\bigg|_0^1=\boxed{\frac1{n+1}}](/tpl/images/0426/3202/23600.png)
![M_X(t)=E[e^{tX}]=\displaystyle\int_{-\infty}^\infty e^{tx}P(X=x)\,\mathrm dx=\int_0^1e^{tx}\,\mathrm dx=\frac{e^{tx}}t\bigg|_0^1=\frac{e^t-1}t](/tpl/images/0426/3202/25813.png)
) is then the 
, so that
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Let x unif[0; 1]. parts (a) and (b) below ask you to nd the nth moment of x in two di erent ways. (...
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