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This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 8. It's a very simple problem, based on bivariate normal distribution, which again teaches us that observing the right thing makes a seemingly laborious problem beautiful . Fun to think, go for it !!

Let have the bivariate normal distribution, ,

where, = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_2\sigma_1 & \sigma^2 \end{pmatrix} ;

Obtain the mean ad variance of .

Bivariate Normal

Conditonal Distribution of Normal

Chi-Squared Distribution

This is a very simple and cute problem, all the labour reduces once you see what to need to see !

Remember , the pdf of ?

Isn't is the exponent of e, in the pdf of bivariate normal ?

So, we can say . Can We ?? verify it !!

Also, clearly ; since follows univariate normal.

So, expectation is easy to find accumulating the above deductions, I'm leaving it as an exercise .

Calculating the variance may be a laborious job at first, but now lets imagine the pdf of the conditional distribution of , what is the exponent of e in this pdf ?? , right !!

and also , . Now doing the last piece of subtle deduction, and claiming that and are independently distributed . Can you argue why ?? go ahead . So, .

So,

, [ since, Variance of a R.V following is .]

Hence the solution concludes.

Before leaving, lets broaden our mind and deal with Multivariate Normal !

Let, be a 1x4 random vector, such that , is positive definite matrix, then can you show that,

Where, .

Keep you thoughts alive !!

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 8. It's a very simple problem, based on bivariate normal distribution, which again teaches us that observing the right thing makes a seemingly laborious problem beautiful . Fun to think, go for it !!

Let have the bivariate normal distribution, ,

where, = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_2\sigma_1 & \sigma^2 \end{pmatrix} ;

Obtain the mean ad variance of .

Bivariate Normal

Conditonal Distribution of Normal

Chi-Squared Distribution

This is a very simple and cute problem, all the labour reduces once you see what to need to see !

Remember , the pdf of ?

Isn't is the exponent of e, in the pdf of bivariate normal ?

So, we can say . Can We ?? verify it !!

Also, clearly ; since follows univariate normal.

So, expectation is easy to find accumulating the above deductions, I'm leaving it as an exercise .

Calculating the variance may be a laborious job at first, but now lets imagine the pdf of the conditional distribution of , what is the exponent of e in this pdf ?? , right !!

and also , . Now doing the last piece of subtle deduction, and claiming that and are independently distributed . Can you argue why ?? go ahead . So, .

So,

, [ since, Variance of a R.V following is .]

Hence the solution concludes.

Before leaving, lets broaden our mind and deal with Multivariate Normal !

Let, be a 1x4 random vector, such that , is positive definite matrix, then can you show that,

Where, .

Keep you thoughts alive !!

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