Respuesta :
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:
[tex]E(X)=30\\SD(X)=4[/tex]
The random variable Y has mean and standard deviation as follows:
[tex]E(Y)=50\\SD(Y)=8[/tex]
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:
[tex]V(Z) = V(35 - 10X)\\=0+10^{2}V(X)\\=100\times (4)^{2}\\=1600[/tex]
Then the standard deviation of Z = 35 - 10X is:
[tex]SD(Z)=\sqrt{V(Z)}\\=\sqrt{1600}\\=40[/tex]
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:
[tex]V(Z) = V(12X-5)\\=12^{2}V(X)+0\\=144\times (4)^{2}\\=2304[/tex]
Then the standard deviation of Z = 12X - 5 is:
[tex]SD(Z)=\sqrt{V(Z)}\\=\sqrt{2304}\\=48[/tex]
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:
[tex]V(Z) = V(X+Y)\\=V(X)+V(Y)+2Cov (X,Y)\\=(4)^{2}+(8)^{2}+0\\=80[/tex]
Then the standard deviation of Z = X + Y is:
[tex]SD(Z)=\sqrt{V(Z)}\\=\sqrt{80}\\=8.94[/tex]
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:
[tex]V(Z) = V(X-Y)\\=V(X)+V(Y)-2Cov (X,Y)\\=(4)^{2}+(8)^{2}+0\\=80[/tex]
Then the standard deviation of Z = X - Y is:
[tex]SD(Z)=\sqrt{V(Z)}\\=\sqrt{80}\\=8.94[/tex]
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:
[tex]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[/tex]
Then the standard deviation of Z = -2X + 2Y is:
[tex]SD(Z)=\sqrt{V(Z)}\\=\sqrt{320}\\=17.89[/tex]
Thus, the variance and standard deviation of Z = -2X + 2Y are 320 and 17.89 respectively.
