Random Variable

Question

Let X and Y are two random variables with E(X)=10,E(Y)=20 and Var(X)=25,Var(Y)=36 and E(X.Y)=225.

Now consider Z=aX-bY+1.

Find a and b when E(Z)=1 and Var(Z)=1.

Comments

Is it a=1/4 and b=1/8?

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don't know the answer. but share your approach brother i am getting a=$\pm1/3$ and b= $\pm1/6$?

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E(Z) = aE(X) - bE(Y) + 1

V(Z) = a^2 V(X) - b^2 V(Y). I used these 2 equation to solve

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@ank73811

I think your Var equation is not right.

$Var(aX-bY)=a^2Var(X)+b^2Var(Y)-2abCov(X,Y)$

$Cov(X,Y)=E(XY)-E(X)E(Y)$ when X and Y are not independent.

$Cov(X,Y)=0$ if X and Y are independent.

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@MiNiPanda, Please confirm in question this should be mentioned that whether random variables are independent or not? Right.

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@Shubhgupta

We can make that out from the given data because if they are independent then Cov(X,Y)=E(XY)-E(X)E(Y) would be 0 but it is 225-200=25 instead which indicates that they are not independent..

Also wanted to know whether V(aX-bY+1) can be written like Var(aX-bY)+Var(1) or not [Var(const)=0]

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Ok yes should be not independent in this case.

Var(aX-bY)+Var(1) or not [Var(const)=0]

I think yes we can write in this form because variance of constant is 0.

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@Shubhgupta

Yes I checked it https://revisionmaths.com/advanced-level-maths-revision/statistics/expectation-and-variance

Thanks :)