The Bivariate Normal Distribution
Worked Example 1: Conditional Distribution
Problem: Suppose (X, Y ) ∼ Bivariate Normal with:
µX = 5, µY = 10, σX = 2, σY = 3, ρ = 0.6.
Find:
label=)The conditional distribution of Y given X = 7.
lbbel=) The mean and variance of Y | X = 7.
Solution:
The conditional distribution formula is:
σY
Y | X = x ∼ N µY + ρ (x − µX ), σY2 (1 − ρ2 ) .
σX
a) Mean:
3
µY |X=7 = 10 + 0.6 · (7 − 5) = 10 + 0.6 · 1.5 · 2 = 10 + 1.8 = 11.8.
2
b) Variance:
σY2 |X=7 = 9 · (1 − 0.36) = 9 · 0.64 = 5.76.
Final Answer:
Y | X = 7 ∼ N (11.8, 5.76).
Sam Maseno University August 19, 2025
This study source was downloaded by 100000889810302 from CourseHero.com on
10-23-2025 06:03:06 GMT -05:00
https://www.coursehero.com/file/252359845/MAS-209-NOTE-pages-5pdf-pages-2pdf/
Worked Example 1: Conditional Distribution
Problem: Suppose (X, Y ) ∼ Bivariate Normal with:
µX = 5, µY = 10, σX = 2, σY = 3, ρ = 0.6.
Find:
label=)The conditional distribution of Y given X = 7.
lbbel=) The mean and variance of Y | X = 7.
Solution:
The conditional distribution formula is:
σY
Y | X = x ∼ N µY + ρ (x − µX ), σY2 (1 − ρ2 ) .
σX
a) Mean:
3
µY |X=7 = 10 + 0.6 · (7 − 5) = 10 + 0.6 · 1.5 · 2 = 10 + 1.8 = 11.8.
2
b) Variance:
σY2 |X=7 = 9 · (1 − 0.36) = 9 · 0.64 = 5.76.
Final Answer:
Y | X = 7 ∼ N (11.8, 5.76).
Sam Maseno University August 19, 2025
This study source was downloaded by 100000889810302 from CourseHero.com on
10-23-2025 06:03:06 GMT -05:00
https://www.coursehero.com/file/252359845/MAS-209-NOTE-pages-5pdf-pages-2pdf/
