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Use the probability distribution given in the table below and consider two new random variables, W= 8 V72Y, to answer the following questions 4x and Joint Distribution of Weather Conditions and Commuting Times Total 23 0.77 1.00 No Rain Long commute (Y = 0) Short commute (Y= 1) Total 0.03 0.22 0.04 0.74 0.78 Compute the mean of W (Round your response to two decimal places) Compute the mean of V. (Round your response to two decimal places) Compute the variance of W (Round your response to four decimal places) Compute the variance of V (Round your response to four decimal places) Compute the covariance between Wand V (Round your response to four decimal places) Compute the correlation between Wand V (Round your response to four decimal places)

Explanation / Answer

from above below is joint probability of (x,y):

marginal density of X:

marginal density of Y:

E(XY) =0.74

hene Cov(X,Y) =E(XY)-E(X)*E(Y)=0.1394

and Correlation (X,Y) =Cov(X,Y)/(Var(X)*Var(Y))1/2 =0.7996

hence:

from above:

E(W) =8+4E(X) =11.12

E(V)=7+2*0.77=8.54

Var(W) =42*Var(X) =2.7456

Var(V)=22*Var(Y)=0.7084

Cov(W,V) =4*2*Cov(X,Y)=1.1152

and Corr(W,V) =0.7996

x y 0 1 Total 0 0.1900 0.0400 0.2300 1 0.0300 0.7400 0.7700 Total 0.2200 0.7800 1.0000

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