1. ## Expectation problem

Allright i got a problem and i actually have the solution i just cant figure some of it out.

Suppose X1,X2,X3 form random sample from uniform distribution on [0,1] determine the value of E[(X1 - 2X2 + X3)^2]

i got it all broken down to

E[x1^2] + 4E[x2^2] + E[x3^2] - 4E[x1]E[x2] + 2E[x1]E[x3] - 4E[x2]E[x3]

the next line in my solution says:
"also since each xi has a uniform distribution on [0,1] E[xi] = 1/2 and
(E[xi^2] = integral 0-1 x^2 dx = 1/3 so desired value of E is 1/2"

I dont get how you get E[xi] is 1/2 and how the fact that integral of exi^2 is 1/3 helps get you to 1/2 any help would be greatly appreciated

2. i figured out the 1/2 part its because its uniformed on [0,1] so E[x] = a+b/2 but i still cant figure out the rest

3. Originally Posted by ChrisBickle
Allright i got a problem and i actually have the solution i just cant figure some of it out.

Suppose X1,X2,X3 form random sample from uniform distribution on [0,1] determine the value of E[(X1 - 2X2 + X3)^2]

i got it all broken down to

E[x1^2] + 4E[x2^2] + E[x3^2] - 4E[x1]E[x2] + 2E[x1]E[x3] - 4E[x2]E[x3]

the next line in my solution says:
"also since each xi has a uniform distribution on [0,1] E[xi] = 1/2 and
(E[xi^2] = integral 0-1 x^2 dx = 1/3 so desired value of E is 1/2"

I dont get how you get E[xi] is 1/2 and how the fact that integral of exi^2 is 1/3 helps get you to 1/2 any help would be greatly appreciated
Your density functin is $f(x)=1$

So by defintion the Expectation is $\mathbb{E}[x]=\int_{0}^{1} xf(x)dx=\int_{0}^{1}xdx=\frac{1}{2}x^2\bigg|_{0}^{ 1}=\frac{1}{2}$

Do the same for $x^2$ to get $\frac{1}{3}$