What does var(X|Y=y) equal in a bivariate normal distribution?

What does var(X|Y=y) equal in a bivariate normal distribution?

• A. σx^2(1 - ρ^2)
• B. σy^2(1 - ρ^2)
• C. σx^2 + σy^2
• D. σx^2(1 + ρ^2)

Answer: (A)

In a bivariate normal distribution, the conditional variance of a normally distributed random variable X given another normally distributed random variable Y, expressed as var(X|Y=y), is influenced by the correlation between the two variables as well as the variance of X itself. The formula for the conditional variance in this context is given by: \[ \text{var}(X|Y=y) = \sigma_x^2(1 - \rho^2) \] Here, σx² represents the variance of the random variable X, and ρ represents the correlation coefficient between X and Y. The term (1 - ρ²) adjusts the variance of X based on the degree of linear relationship between X and Y, effectively reducing the variability of X when Y is observed, particularly when the correlation is high. This answer is correct because it clearly shows how the correlation impacts the conditional variance. When ρ is 1 (perfect correlation), the term (1 - ρ²) becomes 0, indicating that knowing Y provides complete information about X (no uncertainty). Conversely, when ρ is 0 (no correlation), the conditional variance equals the unconditional variance of X, indicating that knowing Y does not provide any information about X. In summary,

In a bivariate normal distribution, the conditional variance of a normally distributed random variable X given another normally distributed random variable Y, expressed as var(X|Y=y), is influenced by the correlation between the two variables as well as the variance of X itself.

The formula for the conditional variance in this context is given by:

[ \text{var}(X|Y=y) = \sigma_x^2(1 - \rho^2) ]

Here, σx² represents the variance of the random variable X, and ρ represents the correlation coefficient between X and Y. The term (1 - ρ²) adjusts the variance of X based on the degree of linear relationship between X and Y, effectively reducing the variability of X when Y is observed, particularly when the correlation is high.

This answer is correct because it clearly shows how the correlation impacts the conditional variance. When ρ is 1 (perfect correlation), the term (1 - ρ²) becomes 0, indicating that knowing Y provides complete information about X (no uncertainty). Conversely, when ρ is 0 (no correlation), the conditional variance equals the unconditional variance of X, indicating that knowing Y does not provide any information about X.

In summary,