Excercise on finding PDF, for Bivariate Guassian.

Review of Formulas


Thus, conditional distribution of Y given X=x is given as,

Example 1

https://online.stat.psu.edu/stat414/lesson/21/21.1
Solution:


Example 2.

Source: https://www.sciencedirect.com/topics/mathematics/bivariate-normal-distribution

Suppose that the heights of fathers and sons are r.v.'s X and Y, respectively, having (approximately) Bivariate Normal distribution with parameters (expressed in inches) μ1 = 70, σ1 = 2, μ2 = 71, σ2 = 2 and ρ = 0.90. If for a given pair (father, son) it is observed that X = x = 69, determine:
(i)

The conditional distribution of the height of the son.

(ii)

The expected height of the son.

(iii)

The probability of the height of the son to be more than 72 in.



Example 2.
determine sigma1, sigma2, correlation coefficient that mu1=0 and mu=-1.
exponent of a bivariate normal density is,




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