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April 19th, 2017, 09:18 PM  #1 
Newbie Joined: Apr 2017 From: India Posts: 23 Thanks: 0  Expectation and Variance
I am looking for the expectation and variance of a random Variable Y whose density function is: f(y)= 1y for 1y<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. 
April 20th, 2017, 03:48 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,763 Thanks: 905 
note that $f(y) = \begin{cases}0 &y< 1 \\1y &1 \leq y \leq 1 \\ 0 &1<y\end{cases}$ 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 $\begin {align*} \displaystyle &\phantom{=}\int_{1}^1 y^2 f(y) \\ \\ &= 2 \int_0^1 ~y^2(1y)~dy \\ \\ &= 2\left(\left . \dfrac 1 3 y^3  \dfrac 1 4 y^4 \right_0^1\right) \\ \\ &= 2\left(\dfrac 1 3  \dfrac 1 4\right) \\ \\ &= 2\left(\dfrac {1}{12}\right) = \dfrac 1 6 \end{align*}$ 

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