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Math Help - Probability and variance

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    Probability and variance

    I don't understand what I missed that I got wrong variance:
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    Quote Originally Posted by totalnewbie View Post
    I don't understand what I missed that I got wrong variance:
    Theorem: Var(aU + b) = a^2 Var(U).

    In your case a = \cos t and b = 0.

    Alternatively:

    E(X) = \int_a^b \frac{u \cos t}{b-a} \, du = \frac{a + b}{2} \, \cos t.

    E(X^2) = \int_a^b \frac{u^2 \cos^2 t}{b-a} \, du = \frac{a^2 + ab + b^2}{3} \, \cos^2 t.

    Note: \frac{b^3 - a^3}{b-a} = \frac{(b-a)(b^2 + ab + a^2)}{b-a} = b^2 + ab + b^2.


    Var(X) = E(X^2) - [E(X)]^2 = \frac{a^2 + ab + b^2}{3} \, \cos^2 t - \frac{(a + b)^2}{4} \, \cos^2 t = \frac{(b-a)^2}{12} \, \cos^2 t.
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