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April 19th, 2017, 08:18 PM   #1
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Expectation and Variance

I am looking for the expectation and variance of a random Variable Y whose density function is:


f(y)= 1-|y| for 1-|y|<1

0 , otherwise

From Here , I have to evaluate the $\int yf(y)$ from $-\infty to \infty$
I am from here. Please help me out.

Answer says Expectation is zero and Variance 1/6.
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April 20th, 2017, 02:48 AM   #2
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note that

$f(y) = \begin{cases}0 &y< -1 \\1-|y| &-1 \leq y \leq 1 \\ 0 &1<y\end{cases}$

This is an even function.

$E[y] = \displaystyle \int_{-1}^1 y f(y)~dy = 0$

because $y f(y)$ is an odd function that is being integrated over a symmetric interval.

similarly, because $y^2 f(y)$ is an even function

$\begin {align*}
\displaystyle

&\phantom{=}\int_{-1}^1 y^2 f(y) \\ \\

&= 2 \int_0^1 ~y^2(1-y)~dy \\ \\

&= 2\left(\left . \dfrac 1 3 y^3 - \dfrac 1 4 y^4 \right|_0^1\right) \\ \\

&= 2\left(\dfrac 1 3 - \dfrac 1 4\right) \\ \\

&= 2\left(\dfrac {1}{12}\right) = \dfrac 1 6

\end{align*}$
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