Question

consider the random variables x and y, representing the annual revenue (in thousands of dollars) at these two restaurants. suppose that x and y have the following properties: mean variance covariance x e(x)=155 var(x)=18 cov(x,y)=8 y e(y)=124 var(y)=26 a. restaurant? has higher expected revenue because? restaurant? may see more varied revenue values because? as the revenue of restaurant x increases, the revenue of restaurant y is likely to? because? b. find the correlation between x and y. (round your answer to 2 decimal places.) c. you decide you would like to invest in both restaurants. define a new variable z = x + y. calculate the expected value, variance, and standard deviation for the combined revenue of restaurants x and y. (round your answers to 2 decimal places.) standard deviation mean variance z = x + y e(z)=279 var(z)=54 σ_z = 7.35

Explanation:

Step1: Calculate expected value of Z

The expected - value of a sum of two random variables is the sum of their expected values. Given \(E(X)=155\) and \(E(Y)=124\), and \(Z = X + Y\), then \(E(Z)=E(X)+E(Y)\).
\[E(Z)=155 + 124=279\]

Step2: Calculate variance of Z

The variance of the sum of two random variables \(X\) and \(Y\) is given by \(Var(Z)=Var(X)+Var(Y)+2Cov(X,Y)\). Given \(Var(X) = 18\), \(Var(Y)=26\), and \(Cov(X,Y)=8\).
\[Var(Z)=18 + 26+2\times8=18 + 26 + 16=60\]

Step3: Calculate standard deviation of Z

The standard deviation \(\sigma_Z\) is the square - root of the variance. So \(\sigma_Z=\sqrt{Var(Z)}\).
\[\sigma_Z=\sqrt{60}\approx7.75\]

Step4: Calculate correlation between X and Y

The formula for the correlation coefficient \(
ho_{XY}=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}\).
\[
ho_{XY}=\frac{8}{\sqrt{18\times26}}=\frac{8}{\sqrt{468}}\approx\frac{8}{21.63}\approx0.37\]

Answer:

Expected value of \(Z\): \(279\)
Variance of \(Z\): \(60\)
Standard deviation of \(Z\): \(7.75\)
Correlation between \(X\) and \(Y\): \(0.37\)