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 !!