Question

use the past below to prove the correlation interval {-1, 1}. For any two variable X and Y we have: Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

Answer 1

we have, to prove that, correlation interval {-1,1}.

for any two variable X and Y , we have,

Var( X+Y) = Var(X)+ Var(Y)+2Cov(X,Y)

ANSWER :-

This is a correlation coefficient,

...............(1)

We prove,

we defined another new random variable which is,

where is unknown.

then variance of new variable is,

where

Variance of Random Variable is always non negative.

we can write that,

..................(2)

by rearrenging terms,

..........(3)

this is bound of covariance values.then we can find the values of from eq.(3), then we diffentiating eq(3) w.r.t. ,

so we have,

put the value of in eq.(3),

Therefore put value of in eq(1)

For lower bound, We get

put in eq(2)

Hence prove,