What is the formula for the conditional expectation of X given Y=y for a bivariate normal distribution?

What is the formula for the conditional expectation of X given Y=y for a bivariate normal distribution?

• A. E[X|Y=y] = μx + ρ(σy/σx)(y - μy)
• B. E[X|Y=y] = μy + ρ(σx/σy)(y - μx)
• C. E[Y|X=x] = μx + ρ(σy/σx)(x - μy)
• D. E[X|Y=y] = μx - ρ(σy/σx)(y - μy)

Answer: (A)

The conditional expectation of a normally distributed random variable, given another normally distributed random variable, has a specific form, which is particularly useful in multivariate settings, such as a bivariate normal distribution. In the case of a bivariate normal distribution with variables X and Y, the formula for the conditional expectation of X given Y equals a particular value y is expressed as: E[X|Y=y] = μx + ρ(σy/σx)(y - μy) In this formula: - μx is the mean of the variable X, - μy is the mean of the variable Y, - ρ is the correlation coefficient between X and Y, - σx is the standard deviation of X, and - σy is the standard deviation of Y. This equation captures how the expected value of X is adjusted based on the observed value of Y (in this case, y). The term (y - μy) represents the deviation of Y from its mean, and it is scaled by the ratio of the standard deviations (σy/σx) and the correlation ρ. This scaling helps to describe how much knowing y influences our expectation of X. The other options either misplace the means, apply the

The conditional expectation of a normally distributed random variable, given another normally distributed random variable, has a specific form, which is particularly useful in multivariate settings, such as a bivariate normal distribution.

In the case of a bivariate normal distribution with variables X and Y, the formula for the conditional expectation of X given Y equals a particular value y is expressed as:

E[X|Y=y] = μx + ρ(σy/σx)(y - μy)

In this formula:

• μx is the mean of the variable X,

• μy is the mean of the variable Y,

• ρ is the correlation coefficient between X and Y,

• σx is the standard deviation of X, and

• σy is the standard deviation of Y.

This equation captures how the expected value of X is adjusted based on the observed value of Y (in this case, y). The term (y - μy) represents the deviation of Y from its mean, and it is scaled by the ratio of the standard deviations (σy/σx) and the correlation ρ. This scaling helps to describe how much knowing y influences our expectation of X.

The other options either misplace the means, apply the