The Bivariate Normal Distribution
Problem: Let (X, Y ) follow a bivariate normal distribution with parameters:
µX = 2, µY = 3, σX = 1, σY = 4, ρ = 0.5.
Find E[Y | X = 3] and Var(Y | X = 3).
label=)
lbbel=)Find the joint PDF of (X, Y ).
Solution:
a) Conditional Mean and Variance:
From the conditional distribution property of the bivariate normal:
σY 2 2
Y | X = x ∼ N µY + ρ (x − µX ), σY (1 − ρ ) .
σX
Step 1: Conditional Mean
4
E[Y | X = 3] = 3 + 0.5 · · (3 − 2) = 3 + 0.5 · 4 · 1 = 3 + 2 = 5.
1
Step 2: Conditional Variance
Var(Y | X = 3) = 42 · (1 − 0.52 ) = 16 · (1 − 0.25) = 16 · 0.75 = 12.
Sam Maseno University August 19, 2025
This study source was downloaded by 100000889810302 from CourseHero.com on
10-23-2025 06:02:45 GMT -05:00
https://www.coursehero.com/file/252359860/MAS-209-NOTE-pages-5pdf-pages-3pdf/
Problem: Let (X, Y ) follow a bivariate normal distribution with parameters:
µX = 2, µY = 3, σX = 1, σY = 4, ρ = 0.5.
Find E[Y | X = 3] and Var(Y | X = 3).
label=)
lbbel=)Find the joint PDF of (X, Y ).
Solution:
a) Conditional Mean and Variance:
From the conditional distribution property of the bivariate normal:
σY 2 2
Y | X = x ∼ N µY + ρ (x − µX ), σY (1 − ρ ) .
σX
Step 1: Conditional Mean
4
E[Y | X = 3] = 3 + 0.5 · · (3 − 2) = 3 + 0.5 · 4 · 1 = 3 + 2 = 5.
1
Step 2: Conditional Variance
Var(Y | X = 3) = 42 · (1 − 0.52 ) = 16 · (1 − 0.25) = 16 · 0.75 = 12.
Sam Maseno University August 19, 2025
This study source was downloaded by 100000889810302 from CourseHero.com on
10-23-2025 06:02:45 GMT -05:00
https://www.coursehero.com/file/252359860/MAS-209-NOTE-pages-5pdf-pages-3pdf/
