Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

Posted by soumen08 on Jun 21, 2014; 5:01am
URL: http://discussion-forum.276.s1.nabble.com/Sample-Questions-of-ISI-ME-I-Mathematics-2010-Discussion-tp4850019p7593243.html

Amit Goyal wrote

Hi Sonal,

a(n) will converge to 0 if the X(i)'s are independent. Here cov(X(i),X(j)) = p and Simple computation will give the following value of variance of a(n),
a(n) = [n(sq(s)) + (sq(n) - n)p]/(sq(n)) = (sq(s)/n) + p - (p/n)
Hence, lim(n→∞) a(n) = p

Are you sure on this one? You seem to be thinking of X(i) as numbers. They are distributions (meaning that they are a sequence of numbers). Intuitively, when you add two distributions, the variance of the sum must be greater than their individual variances. The formula is actually -

Var(A+B) = Var(A) + Var(B) + 2*Cov(A,B)

Extending, to n terms, we will get

Var(A1+A2+....An) = n*sigma^2 + n*rho

But we have 1/n outside. So, we get Sigma^2+rho which is option (d)

I'd like someone to confirm or deny this. Am I right? Am I missing something?