Suppose a restaurant in town serves three fixed price dinners costing $12, $15, and $20. For a
randomly selected couple (
Penny
= X and
Leonard=
Y), the joint probability table for their selections
is given below.
Y
p(x,y)
12
15
20
12
.05
.05
.10
15
.05
.10
.35
20
0
.20
.10
a.
Find the marginal distributions of X and Y.
b.
Find Cov(X,Y) and Corr(X,Y). What is the strength
and direction of the correlation?
c.
Find the conditional distribution of X given Y=12.
d.
Find E(X|
Y=12) and Var(X|Y=12)
a. To find the marginal distributions of X and Y, we need to sum the probabilities over the rows and columns of the joint probability table, respectively.
The marginal distribution of X (P(X)) can be found by summing the probabilities in each row of the table:
P(X=12) = 0.05 + 0.05 + 0.10 = 0.20
P(X=15) = 0.05 + 0.10 + 0.35 = 0.50
P(X=20) = 0 + 0.20 + 0.10 = 0.30
The marginal distribution of Y (P(Y)) can be found by summing the probabilities in each column of the table:
P(Y=12) = 0.05 + 0.05 + 0 = 0.10
P(Y=15) = 0.05 + 0.10 + 0.20 = 0.35
P(Y=20) = 0.10 + 0.35 + 0.10 = 0.55
b. To find Cov(X, Y) and Corr(X, Y), we need to calculate the means and variances of X and Y, as well as the covariance between X and Y.
Mean of X (μX) = Σ(X * P(X))
= (12 * 0.20) + (15 * 0.50) + (20 * 0.30) = 15.70
Mean of Y (μY) = Σ(Y * P(Y))
= (12 * 0.10) + (15 * 0.35) + (20 * 0.55) = 17.15
Variance of X (Var(X)) = Σ((X - μX)^2 * P(X))
= ((12 - 15.70)^2 * 0.20) + ((15 - 15.70)^2 * 0.50) + ((20 - 15.70)^2 * 0.30) = 4.195
Variance of Y (Var(Y)) = Σ((Y - μY)^2 * P(Y))
= ((12 - 17.15)^2 * 0.10) + ((15 - 17.15)^2 * 0.35) + ((20 - 17.15)^2 * 0.55) = 8.4775
Covariance of X and Y (Cov(X, Y)) = Σ((X - μX) * (Y - μY) * P(X, Y))
= ((12 - 15.70) * (12 - 17.15) * 0.05) + ((12 - 15.70) * (15 - 17.15) * 0.05) + ((12 - 15.70) * (20 - 17.15) * 0.10)
+ ((15 - 15.70) * (12 - 17.15) * 0.05) + ((15 - 15.70) * (15 - 17.15) * 0.10) + ((15 - 15.70) * (20 - 17.15) * 0.35)
+ ((20 - 15.70) * (12 - 17.15) * 0) + ((20 - 15.70) * (15 - 17.15) * 0.20) + ((20 - 15.70) * (20 - 17.15) * 0.10)
Correlation coefficient of X and Y (Corr(X, Y)) = Cov(X, Y) / √(Var(X) * Var(Y))
After calculating the above expressions, you can find the Cov(X, Y) and Corr(X, Y).
c. To find the conditional distribution of X given Y = 12 (P(X|Y=12)), we need to calculate the probabilities of X when Y is fixed at 12.
P(X=12|Y=12) = 0.05 / P(Y=12) = 0.05 / 0.10 = 0.50
P(X=15|Y=12) = 0.05 / P(Y=12) = 0.05 / 0.10 = 0.50
P(X=20|Y=12) = 0.10 / P(Y=12) = 0.10 / 0.10 = 1
d. To find E(X|Y=12) (the conditional expectation of X given Y = 12), we need to calculate the expected value of X when Y is fixed at 12.
E(X|Y=12) = Σ(X * P(X|Y=12))
= (12 * 0.50) + (15 * 0.50) + (20 * 1) = 22.5
To find Var(X|Y=12) (the conditional variance of X given Y = 12), we need to calculate the variance of X when Y is fixed at 12.
Var(X|Y=12) = Σ((X - E(X|Y=12))^2 * P(X|Y=12))
= ((12 - 22.5)^2 * 0.50) + ((15 - 22.5)^2 * 0.50) + ((20 - 22.5)^2 * 1)