Question

Suppose X is an exponential random variable with mean 5 and Y is an exponential random variable with mean 10. X and Y are independent. Determine the coefficient of variation of X + Y

Answer 1

It is given that X is an exponential random variable with mean 5 and Y is an exponential random variable with mean 10. X and Y are independent.

Consider the density of exponential variate  X~ Exp()  be

Then E(x)=  

E(X²)=

V(x)=

Coefficient of variation (CV) of (X+Y)= Standard deviation(X+Y)*100/Mean(X+Y) ............ Equation A

Let us consider that in our case

then using previous results we have

E(x)=   

Using  

Similary E(Y)=

Hence V(Y)=100

Now, mean(X+Y)=E(X+Y)=E(X)+E(Y)=5+10=15

Var(X+Y)=Var(X)+Var(Y) = 25+100  ( As X and Y are independent)

SD(X+Y)= =11.180

Using above results in Equation , we have

Hence CV(X+Y)=11.18*100/15=74.53

Hence coefficient of variation (X+Y)=74.53