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Math Help - Expectation problem

  1. #1
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    Post 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
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  2. #2
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    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
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  3. #3
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    Quote Originally Posted by ChrisBickle View Post
    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}
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