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Math Help - [SOLVED] mean and variance

  1. #1
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    [SOLVED] mean and variance

    Let X1,X2,X3,X4 be four iid random variables having the same pdf f(x)=2x, 0<x<1, zero elsewhere. Find the mean and variance of the sum Y of these four random variables. ??
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ninano1205 View Post
    Let X1,X2,X3,X4 be four iid random variables having the same pdf f(x)=2x, 0<x<1, zero elsewhere. Find the mean and variance of the sum Y of these four random variables. ??
    I think you mean to say that Y=X_1+X_2+X_3+X_4?

    Thus, E\left[Y\right]=E\left[X_1+X_2+X_3+X_4\right]=E\left[X_1\right]+E\left[X_2\right]+E\left[X_3\right]+E\left[X_4\right]

    Since they all have the same pdf, we can say that E\left[Y\right]=4\int_0^1 \left[x\cdot\left(2x\right)\right]\,dx=8\int_0^1 x^2\,dx. Can you take it from here?

    For variances, \text{Var}\left[Y\right]=\text{Var}\left[X_1+X_2+X_3+X_4\right]=\text{Var}\left[X_1\right]+\text{Var}\left[X_2\right]+\text{Var}\left[X_3\right]+\text{Var}\left[X_4\right]

    Since they all have the same pdf, we can say that \text{Var}\left[Y\right]=4\int_0^1\left[x^2\cdot\left(2x\right)\right]\,dx=8\int_0^1 x^3\,dx

    Can you take it from here? Does this make sense?
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  3. #3
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    Variance?

    I got the mean = 8/3 which is right but the variance doesnt seem to be right. Following ur way the var comes out to be 2 but the answer should be 2/9. What's wrong with it?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ninano1205 View Post
    I got the mean = 8/3 which is right but the variance doesnt seem to be right. Following ur way the var comes out to be 2 but the answer should be 2/9. What's wrong with it?
    Woops...I forgot something...

    Var\left[x\right]=E\left[x^2\right]-{\color{red}\left(E\left[x\right]\right)^2}

    So you would need to evaluate 4\left[\int_0^1 2x^3\,dx-\left(\int_0^1 2x^2\,dx\right)^2\right] to get the variance (I multiplied the value by 4 because each r.v. has the same variance). It should be \frac{2}{9}

    Does this make sense?
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  5. #5
    MHF Contributor matheagle's Avatar
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    That's right.
    You found the second moment and not the variance.
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