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finding distribution from multivariate Gaussian

Let Z1,Z2,Z3,Z4Z_1,Z_2,Z_3,Z_4 be independent N(0,σ\sigma^2) random variables and let Y=(Y1,Y2,Y3,Y4)TY=(Y_1,Y_2,Y_3,Y_4)^T be a random vector defined by Y=BZY=BZ where B=(1111111111111111)\begin{pmatrix} 1 & 1 & 1 & 1 \\1 & -1 & -1 & 1 \\1 & -1 & 1 & -1 \\1 & 1 & -1 & -1 \end{pmatrix}

Determine the distribution of Y1+Y2sqrt(Y32+Y42) \frac{Y_1 + Y_2}{sqrt{(Y_3^2 + Y_4^2)}} and (Y3+Y4)2Y12+Y22 \frac{(Y_3 + Y_4)^2}{Y_1^2 + Y_2^2}

I started off by finding the distibution of Y. We know that the Z random variables are independent and if we let Z=(Z1,Z2,Z3,Z4)TZ=(Z_1,Z_2,Z_3,Z_4)^T, the mean would be the 0 vector and variance-covariance would be the identity matrix. Therefore, the mean of Y would be the 0 vector and the variance-covariance would be BYBT=4IB\sum_Y B^T = 4I.

The distributions I have to find must be Gaussian also, and have a mean of 0. But i'm not sure how to work out the variance.
Any help would be appreciated
Original post by ellemay96
Let Z1,Z2,Z3,Z4Z_1,Z_2,Z_3,Z_4 be independent N(0,σ\sigma^2) random variables and let Y=(Y1,Y2,Y3,Y4)TY=(Y_1,Y_2,Y_3,Y_4)^T be a random vector defined by Y=BZY=BZ where B=(1111111111111111)\begin{pmatrix} 1 & 1 & 1 & 1 \\1 & -1 & -1 & 1 \\1 & -1 & 1 & -1 \\1 & 1 & -1 & -1 \end{pmatrix}

Determine the distribution of Y1+Y2sqrt(Y32+Y42) \frac{Y_1 + Y_2}{sqrt{(Y_3^2 + Y_4^2)}} and (Y3+Y4)2Y12+Y22 \frac{(Y_3 + Y_4)^2}{Y_1^2 + Y_2^2}

I started off by finding the distibution of Y. We know that the Z random variables are independent and if we let Z=(Z1,Z2,Z3,Z4)TZ=(Z_1,Z_2,Z_3,Z_4)^T, the mean would be the 0 vector and variance-covariance would be the identity matrix. Therefore, the mean of Y would be the 0 vector and the variance-covariance would be BYBT=4IB\sum_Y B^T = 4I.


OK, you've found that the Y's are uncorrelated and therefore (because they're normal RVs) they are independent. That really helps in the next bit!

The distributions I have to find must be Gaussian also, and have a mean of 0. But i'm not sure how to work out the variance.
Any help would be appreciated


Let

A1=Y1+Y2(Y32+Y42) \displaystyle A1 = \frac{Y_1 + Y_2}{\sqrt{(Y_3^2 + Y_4^2)}}

and

A2=(Y3+Y4)2Y12+Y22 \displaystyle A2 = \frac{(Y_3 + Y_4)^2}{Y_1^2 + Y_2^2}


Look at A2A2: it's something squared over a sum of squared things - it's always going to be positive. It can't be normally distributed. So, if XX is normally distributed, then what's the distribution of X2X^2? What sort of random variable do you get when you take a ratio of two independent variables with this latter distribution? Similarly for A2A2: can you think of the definition of a common-or-garden probability distribution that might be relevant here.

If you have R available, you might like to play with this code so get your intuition working. Draw some plots!

N <- 100000

Z1 <- rnorm(N)
Z2 <- rnorm(N)
Z3 <- rnorm(N)
Z4 <- rnorm(N)

Y1 <- Z1 + Z2 + Z3 + Z4
Y2 <- Z1 - Z2 - Z3 + Z4
Y3 <- Z1 - Z2 + Z3 - Z4
Y4 <- Z1 + Z2 - Z3 - Z4

A1 <- (Y1 + Y2)/sqrt(Y3^2 + Y4^2)
A2 <- ((Y3 + Y4)^2)/(Y1^2 + Y2^2)
(edited 8 years ago)
Reply 2
Original post by Gregorius
OK, you've found that the Y's are uncorrelated and therefore (because they're normal RVs) they are independent. That really helps in the next bit!



Let

A1=Y1+Y2(Y32+Y42) \displaystyle A1 = \frac{Y_1 + Y_2}{\sqrt{(Y_3^2 + Y_4^2)}}

and

A2=(Y3+Y4)2Y12+Y22 \displaystyle A2 = \frac{(Y_3 + Y_4)^2}{Y_1^2 + Y_2^2}


Look at A2A2: it's something squared over a sum of squared things - it's always going to be positive. It can't be normally distributed. So, if XX is normally distributed, then what's the distribution of X2X^2? What sort of random variable do you get when you take a ratio of two independent variables with this latter distribution? Similarly for A2A2: can you think of the definition of a common-or-garden probability distribution that might be relevant here.

If you have R available, you might like to play with this code so get your intuition working. Draw some plots!

N <- 100000

Z1 <- rnorm(N)
Z2 <- rnorm(N)
Z3 <- rnorm(N)
Z4 <- rnorm(N)

Y1 <- Z1 + Z2 + Z3 + Z4
Y2 <- Z1 - Z2 - Z3 + Z4
Y3 <- Z1 - Z2 + Z3 - Z4
Y4 <- Z1 + Z2 - Z3 - Z4

A1 <- (Y1 + Y2)/sqrt(Y3^2 + Y4^2)
A2 <- ((Y3 + Y4)^2)/(Y1^2 + Y2^2)


Thank you! I completely overlooked other distributions. so does that mean A_1 has a t distribution and A_2 has chi squared? How would you go about finding the parameters?
Original post by ellemay96
Thank you! I completely overlooked other distributions. so does that mean A_1 has a t distribution and A_2 has chi squared? How would you go about finding the parameters?


A1 is the ratio of a normal distribution and the square root of a Chi-squared distribution on <how many?> degrees of freedom and therefore has something like a t-distribution on that number of df.

A2 is a ratio of a Chi-squared on 1df with a Chi-squared on 2df and therefore has a <which distribution? - not chi-squared!> on what pair of dfs?

take a look at the definition of the distributions we're aiming at and take note that we may not have normalized them appropriately to get "unit" distributions.
Reply 4
Original post by Gregorius
A1 is the ratio of a normal distribution and the square root of a Chi-squared distribution on <how many?> degrees of freedom and therefore has something like a t-distribution on that number of df.

A2 is a ratio of a Chi-squared on 1df with a Chi-squared on 2df and therefore has a <which distribution? - not chi-squared!> on what pair of dfs?

take a look at the definition of the distributions we're aiming at and take note that we may not have normalized them appropriately to get "unit" distributions.


Okay, i'll read over my notes again. Thank you so much for your help!

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