Question

Suppose that X is a random variable with mean 30 and standard deviation 4. Also suppose that Y is a random variable with mean 50 and standard deviation of 8. Assume that the correlation between X and Y is zero.
1. Find the variance and the standard deviation of the random variable Z for each of the following cases. Show work.
(a) Z = 35 -10X
(b) Z = 12X - 5
(c) Z = X +Y
(d) Z = X -Y
(e) Z = -2X + 2Y

Answer 1

Answer:

(a) The variance and standard deviation of Z = 35 - 10X are 1600 and 40 respectively.

(b) The variance and standard deviation of Z = 12X - 5 are 2304 and 48 respectively.

(c) The variance and standard deviation of Z = X + Y are 80 and 8.94 respectively.

(d) The variance and standard deviation of Z = X - Y are 80 and 8.94 respectively.

(e) The variance and standard deviation of Z = -2X + 2Y are 320 and 17.89 respectively.

Step-by-step explanation:

The random variable X has mean and standard deviation as follows:

$E(X)=30\\SD(X)=4$

The random variable Y has mean and standard deviation as follows:

$E(Y)=50\\SD(Y)=8$

It is provided that the correlation between X and Y is 0.

This implies that Cov (X, Y) = 0.

(a)

Compute the variance of Z = 35 - 10X as follows:

$V(Z) = V(35 - 10X)\\=0+10^{2}V(X)\\=100\times (4)^{2}\\=1600$

Then the standard deviation of Z = 35 - 10X is:

$SD(Z)=\sqrt{V(Z)}\\=\sqrt{1600}\\=40$

Thus, the variance and standard deviation of Z = 35 - 10X are 1600 and 40 respectively.

(b)

Compute the variance of Z = 12X - 5 as follows:

$V(Z) = V(12X-5)\\=12^{2}V(X)+0\\=144\times (4)^{2}\\=2304$

Then the standard deviation of Z = 12X - 5 is:

$SD(Z)=\sqrt{V(Z)}\\=\sqrt{2304}\\=48$

Thus, the variance and standard deviation of Z = 12X - 5 are 2304 and 48 respectively.

(c)

Compute the variance of Z = X + Y as follows:

$V(Z) = V(X+Y)\\=V(X)+V(Y)+2Cov (X,Y)\\=(4)^{2}+(8)^{2}+0\\=80$

Then the standard deviation of Z = X + Y is:

$SD(Z)=\sqrt{V(Z)}\\=\sqrt{80}\\=8.94$

Thus, the variance and standard deviation of Z = X + Y are 80 and 8.94 respectively.

(d)

Compute the variance of Z = X - Y as follows:

$V(Z) = V(X-Y)\\=V(X)+V(Y)-2Cov (X,Y)\\=(4)^{2}+(8)^{2}+0\\=80$

Then the standard deviation of Z = X - Y is:

$SD(Z)=\sqrt{V(Z)}\\=\sqrt{80}\\=8.94$

Thus, the variance and standard deviation of Z = X - Y are 80 and 8.94 respectively.

(e)

Compute the variance of Z = -2X + 2Y as follows:

$V(Z) = V(-2X+2Y)\\=(-2)^{2}V(X)+(2)^{2}V(Y)+2(-2)(2)Cov (X,Y)\\=4\times(4)^{2}+4\times(8)^{2}+0\\=320$

Then the standard deviation of Z = -2X + 2Y is:

$SD(Z)=\sqrt{V(Z)}\\=\sqrt{320}\\=17.89$

Thus, the variance and standard deviation of Z = -2X + 2Y are 320 and 17.89 respectively.