Suppose that X is a random variable with mean 15 and standar

Suppose that X is a random variable with mean 15 and standard deviation 5. Also suppose that Y is a random variable with mean 20 and standard deviation 10. X and Y are independent; find out the variance and standard deviation of the random variable Z for each of the following cases. Show your work.

(a) Z=2+10X

(b) Z=X+Y

What if the correlation between X and Y is 0.5 and all the other conditions remain the same? What is the mean and standard deviation of Z in (a)(b) then?

Solution

a) Z =2 + 10X ,

so each value is multiplied by 10 and added 2 to it

it can be seen that the pattern exist for mean also

hence

mean = 15 * 10 + 2 = 152

variance will not change each will still is at same distance from mean

=>

standard deviation = 5

b)

Z=X+Y

mean = u1+u2 / 2 = 15 + 20 /2 = 17.5

standard deviation = sqrt( std(X)^2 + std(Y)^2)

= sqrt(25 + 100)

= 5 * sqrt(5)

c)

the values of a) doesnot change as it doesnot depend on the both the variables

the mean of (b) will not change as correlation doesnot effect mean

but

standard deviation = sqrt ( std(X)^2 + std(Y)^2 + 2 * corr * std(X) * std(Y))

= sqrt(5^2 + 10^2 + 2 * 0.5 * 5 * 10)

= 5 * sqrt(7)