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How to show that if X is a nonnegative integervalued random variable with distribution F,then

$$E(X)=\displaystyle\int_0^\infty \overline{F}(X)dx$$

and

$$E(X^n)=\displaystyle\int_0^\infty n*X^{n-1}\overline{F}(X)dx$$

where $F(X)=P(X\leq x)$ and $\overline{F}(X)=(X > x)$.

My attempt: I am working on this question. Meanwhile if any member knows the answer to this question may reply with correct answer.

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    $\begingroup$ Your first equation holds for any nonnegative random variable, not just for those that take on integer values only. For nonnegative integer-valued random variables, one can replace the integral by a sum. $\endgroup$ May 23, 2020 at 15:59
  • $\begingroup$ Sorry, I did not get the definition of $\overline F (X)$. $\endgroup$
    – Pohoua
    May 23, 2020 at 18:42
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    $\begingroup$ See math.stackexchange.com/q/172841/321264. $\endgroup$ May 23, 2020 at 20:01

1 Answer 1

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For integer $n \geq 0$, suppose that $X$ takes on value $n$ with probability $p_n$ where the $p_n$'s are nonnegative numbers such that $\sum_{n=0}^\infty p_n = 1$. Note that we don't assume that all the $p_n$ are nonzero; in fact, in many cases of interest, $p_n = 0$ for all $n > M$ for some finite integer $M$. Be that as it may, notice that the graph of $1-F(x)$ is a descending staircase function having value $1$ for $x<0$, and value $1-p_0$ for $x\in [0,1)$, value $1-p_0-p_1$ for $x\in [1,2)$, value $1-p_0-p_1-p_2$ for $x\in [2,3)$, and so on. So, the area under the $1-F(x)$ staircase on the positive axis is the sum of the areas of rectangles of base $1$ and heights $$\begin{array}{lclclclclcl}1-p_0 &= &p_1&+&p_2&+&p_3&+&p_4&+&\cdots\\ 1-p_0-p_1&= &&&p_2&+&p_3&+&p_4&+&\cdots\\ 1-p_0-p_1-p_2&= &&&&&p_3&+&p_4&+&\cdots\\ \vdots \qquad \ddots & \vdots \end{array}$$ Instead of adding by rows in the above table, if we add by columns to calculate the total area under $1-F(x)$, we see that $$\int_0^\infty [1-F(x)]\,\mathrm dx = \sum_{n=1}^\infty n\cdot p_n = \sum_{n=0}^\infty n\cdot p_n = E[X].$$

So if we are agreed that $E[X]=\int_0^\infty P(X>x)\,\mathrm dx$ holds for any nonnegative random variable (not just integer-valued ones), it holds for $Y=X^n$ also and so we have \begin{align} E[Y] &= \int_0^\infty P(Y>y)\,\mathrm dy\\ &= \int_0^\infty P(X^n>y)\,\mathrm dy\\ &= \int_0^\infty P(X>y^{\frac 1n})\,\mathrm dy. \end{align} Now make a change of variables in that last integral by setting $y^{\frac 1n} = x, \frac 1n y^{\frac 1n-1}\,\mathrm dy = \mathrm dx$, that is, $\mathrm dy = n x^{n-1}\mathrm dx$ to get $$E[X^n] = \int_0^\infty n x^{n-1}P(X>x) \,\mathrm dx.$$

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  • $\begingroup$ I know second equality is also true but i cann't state it in a mathematical language step by step. Do you know how to prove the second equality? $\endgroup$ May 24, 2020 at 13:24

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