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Math Help - Correlation and Lognormality

  1. #1
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    Correlation and Lognormality

    Hi everyone.

    At the moment, im just doing some probability equations which i am not too 100% sure about. Been trying to get around things to solve it but out of luck. But if anyone can help that would be wonderful! Assisting me in either one of my problems is highly appreciated. Thank you!

    ok first question:

    Let E(X) = 20, E(Y) = 10, Var(X) = 4 and Var(Y) = 1 and p=0.5 Additionally assume they are independent Find the covariance and correlation between X and U = 2X - 3Y.


    And last question. Let X1 and X2 have both lognormal LN(0,1) and are both independent. Find the distribution of: X1/X2 and P(X1 > 2X2)

    Hope you can gimme a hand on this one. Again thanks again
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  2. #2
    Moo
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    Hello,
    Let E(X) = 20, E(Y) = 10, Var(X) = 4 and Var(Y) = 1 and p=0.5 Additionally assume they are independent Find the covariance and correlation between X and U = 2X - 3Y.
    Using the bilinearity of the covariance, we have cov(X,U)=cov(X,2X-3Y)=2cov(X,X)-3cov(X,Y)
    Since X and Y are independent, cov(X,Y)=0
    And cov(X,X)=Var(X)
    You're done for this one

    As for the correlation, you'll need Var(U), which is Var(2X-3Y)=4Var(X)+9Var(Y) (because they're independent)
    You're done for this one too

    And last question. Let X1 and X2 have both lognormal LN(0,1) and are both independent. Find the distribution of: X1/X2 and P(X1 > 2X2)
    For the distribution of X1/X2, let's do it a smart way
    Since they're lognormal, there exist Y1 and Y2 both N(0,1) such that X1=exp(Y1) and X2=exp(Y2), and which are independent.
    For further convenience, we can actually let X2=exp(-Y2) (it doesn't change much, because -N(0,1)=N(0,1))

    So X1/X2=exp(Y1+Y2)

    Since Y1 and Y2 are independent, Y1+Y2 follows a normal distribution (0,2)
    So this is a lognormal (0,2)

    As for P(X1>2X2), note that it equals P(X1/X2>2)... or if you want to relate to a normal distribution, it equals P(N(0,2)>ln(2)).

    If something I wrote is unclear, please tell me. It's late, and I was a bit lazy to explain very clearly
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

    Using the bilinearity of the covariance, we have cov(X,U)=cov(X,2X-3Y)=2cov(X,X)-3cov(X,Y)
    Since X and Y are independent, cov(X,Y)=0
    And cov(X,X)=Var(X)
    You're done for this one

    As for the correlation, you'll need Var(U), which is Var(2X-3Y)=4Var(X)+9Var(Y) (because they're independent)
    You're done for this one too


    For the distribution of X1/X2, let's do it a smart way
    Since they're lognormal, there exist Y1 and Y2 both N(0,1) such that X1=exp(Y1) and X2=exp(Y2), and which are independent.
    For further convenience, we can actually let X2=exp(-Y2) (it doesn't change much, because -N(0,1)=N(0,1))

    So X1/X2=exp(Y1+Y2)

    Since Y1 and Y2 are independent, Y1+Y2 follows a normal distribution (0,2)
    So this is a lognormal (0,2)

    As for P(X1>2X2), note that it equals P(X1/X2>2)... or if you want to relate to a normal distribution, it equals P(N(0,2)>ln(2)).

    If something I wrote is unclear, please tell me. It's late, and I was a bit lazy to explain very clearly
    thanks for your reply Moo! Much appreciated.

    So for the first question could you briefly explain your approach about the covariance bit where you took the constants out. It seems like my teachers were a bit lazy in doing so. and why did the 2 disappeared when it came to cov(x,X) = Var(X)?

    And how would you put in the values (the numbers) for the first question? :3

    And the last question it should be:

    P((X - u)/2- (ln(2) - 0/ 2)) where 2 in the denominator = variance so the final answer is:

    P(Z > Ln(2)/2))?

    Thank you so much for your reply
    Last edited by Redeemer_Pie; March 28th 2010 at 05:51 PM.
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  4. #4
    Moo
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    Quote Originally Posted by Redeemer_Pie View Post
    thanks for your reply Moo! Much appreciated.

    So for the first question could you briefly explain your approach about the covariance bit where you took the constants out. It seems like my teachers were a bit lazy in doing so.
    Hmmm... We have cov(X,Y)=E[XY]-E[X]E[Y]
    Everything can come from here ! Try it, it's not difficult at all !

    and why did the 2 disappeared when it came to cov(x,X) = Var(X)?
    It didn't, I just put that cov(X,X)=var(X), then you indeed need 2cov(X,X). That's all.

    And how would you put in the values (the numbers) for the first question? :3
    For the covariance, everything has been given in the problem you know the variance of X.
    As for the correlation, just use the standard definition.

    And the last question it should be:

    P((X - u)/2- (ln(2) - 0/ 2)) where 2 in the denominator = variance so the final answer is:

    P(Z > Ln(2)/2))?
    That's why I put P(N(0,2)>ln(2)), meaning that it's P(T>ln(2)) where T ~ N(0,2)

    As for getting a standard normal distribuiton, you have to divide by sqrt(2), and not 2

    Thank you so much for your reply
    You're welcome. I hope it helps !
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