Question

Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Assume that the correlation between X and Y is 0.5. Find the mean and variance of the random variable Z = ????3X ???? 2Y . Be sure to show all your work.

Answer 1

a

`E(Z) =  202`

b

`E(Z) = 198`

c

`E(Z) = -140`

Explanation:

From the question we are told that

The mean of X is
`E(X)  =  20`

The standard deviation of X is
`s_1 = 5`

The mean of Y is
`\= y  =  40`

The standard deviation of Y is
`s_2 = 10`

Considering question a

Generally the mean of Z = 2 + 10X. is mathematically represented

`E(Z) = E[2 + 10X ]`

=>
`E(Z) =  2 +  10 E(X)`

=>
`E(Z) =  2 +  10 * 20`

=>
`E(Z) =  202`

Considering question b

Generally the mean of Z = 10X - 2.. is mathematically represented

`E(Z) = E[10X -2 ]`

=>
`E(Z) = 10E(X) - 2`

=>
`E(Z) = 10* 20  - 2`

=>
`E(Z) = 200  - 2`

=>
`E(Z) = 198`

Considering question c

Generally the mean of -3X - 2Y is mathematically represented

`E(Z) = E[-3X -2Y ]`

`E(Z) = -3 E(X) -2E(Y)`

=>
`E(Z) = -3 * 20 -2* 40`

=>
`E(Z) = -140`