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Write down joint probability density function $f_{XY}(x,y)$ of two RVs $X$ and $Y$ given $X = Z_2 - Z_3$ and $Y = Z_1 - Z_2 + Z_3$ where $Z_1$, $Z_2$ and $Z_3$ are independent normal random variables each having mean $0$ and variance $1$.

I tried using change of variable formula but couldn't find determinant because it wasn't an $N\times N$ matrix.

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closed as off-topic by Juho Kokkala, kjetil b halvorsen, Michael Chernick, John, Peter Flom May 18 '17 at 11:30

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  • $\begingroup$ You might need to add the self-study tag as this is a homework like problem. You should know that X and Y are marginally normal. Why not calculate the variances and the covariance for starters. They both will of course have zero mean. $\endgroup$ – Michael Chernick May 17 '17 at 18:51
  • $\begingroup$ I calculated the cov(X,Y) and the variances of both which allowed me to get corr(X,Y). Its not independent so I could not use the fact f(x)f(y)=f(xy). I found a formula for the bivariate case. I'll link the page. Could you tell me if the formula is correct to use because I did this question another way and I'm getting different answers. Thanks in advance. en.wikipedia.org/wiki/Multivariate_normal_distribution $\endgroup$ – Todd May 18 '17 at 14:36
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Method 1: Use the fact that lineal combination of normal random variable is still normal. In matrix, we have $AZ$ ~ $N(A(E(Z)), A(Var(Z))A')$ given $Z$ is noraml.

Method 2: If homework asks you to use determinant method, you can add another component: such as $Z = Z_1 -Z_2 -2Z_3$. Then find the joint distribution of $(X, Y, Z)$ using determinant method. One more step is $\int f(x,y,z)dz$ to get pdf of $(X, Y)$.

Method 3: Characteristic function

Method 4: ? I do not know.

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  • $\begingroup$ I would say that you could calculate the covariance directly since you know that $Z_1$, $Z_2$ and $Z_3$ are all independent standard normal random variables. You may need to make an argument why (X,Y) is bivariate normal. Then you can write down the joint density without doing any integration. $\endgroup$ – Michael Chernick May 17 '17 at 20:20
  • $\begingroup$ You are right. But I think method 2 is what teacher wanted. $\endgroup$ – user158565 May 18 '17 at 14:21
  • $\begingroup$ Thanks guys. How would you find the marginal pdf of f(y) without evaluating an interval given your answer to the first part $\endgroup$ – Todd May 18 '17 at 14:27
  • $\begingroup$ See fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node7.html Marginal distribution of Y is still normal distribution and the mean and variance have no change. $\endgroup$ – user158565 May 18 '17 at 15:02

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