Let N,X1,Y1,X2,Y2,… be independent random variables. The random variable N takes positive integer values and has mean a and variance r. The random variables Xi are independent and identically distributed with mean b and variance s, and the random variables Yi are independent and identically distributed with mean c and variance t. Let
A=∑i=1NXi and B=∑i=1NYi.
Find cov(A,B). Express your answer in terms of the given means and variances using standard notation.
cov(A,B)=?
Find var(A+B). Express your answer in terms of the given means and variances using standard notation.
var(A+B)=?
8 answers
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cov(A,B) = b*c*r
var(A+B) = a*(s+t)+r*(c+b)^2
var(A+B) = a*(s+t)+r*(c+b)^2
For var(a+b), you need to include 2 x the covariance
why the convariance need to be multiplied by 2? can anyone elucidate this part?
Because the variance of the sum of two RV's is only equal to the sum of the variances if the two RV's are independent. If the two RV's are dependent then the variance of the sum of the two RV's = sum of the variances + 2*COV(the two RV's)
cov(A,B) = b*c*r
var(A+B) = a*(s+t)+r*(c^2+b^2)+2*b*c*r
var(A+B) = a*(s+t)+r*(c^2+b^2)+2*b*c*r
0 answers