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Math Help - Probability Bivariate RV Mean and Variance

  1. #1
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    Probability Bivariate RV Mean and Variance

    Let X and Y be independent random variables with E(X)=E(Y)=5, var(X)=1 and var(Y)=(sigma^2)>1.

    Put Z=aX + (1-a)Y, 0<=a<=1.

    Find 1. the value of a that minimises var(A) and that minimum value.

    2. the value of a that maximises var(Z) and that maximum value.

    Help!
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  2. #2
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    Quote Originally Posted by maibs89 View Post
    Let X and Y be independent random variables with E(X)=E(Y)=5, var(X)=1 and var(Y)=(sigma^2)>1.

    Put Z=aX + (1-a)Y, 0<=a<=1.

    Find 1. the value of a that minimises var(A) and that minimum value. Mr F says: What is the definition of the random variable A?

    2. the value of a that maximises var(Z) and that maximum value.

    Help!
    Since X and Y are independent you can say that

    Var(Z) = f(a) = a Var(X) + (1 - a) Var(Y).

    Substitute for Var(X) and Var(Y) and

    1. Minimise f over 0<=a<=1 ....... (assuming that A is a typo and you meant Z).

    2. Maximise f over 0<=a<=1 .......

    Note that 1 - \sigma^2 < 0 .......
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  3. #3
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    a is a constant.

    I worked my way until Var(Z) = f(a) = a Var(X) + (1 - a) Var(Y).

    Yes, I made a typo.

    How do I find Var(Y)?
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  4. #4
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    Quote Originally Posted by maibs89 View Post
    [snip]
    How do I find Var(Y)?
    You read the question: "....... var(Y)=(sigma^2)>1."
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