Conditional expectations of bivariate normal distributions

In summary, a bivariate normal distribution is a type of multivariate normal distribution that describes the relationship between two continuous random variables. Conditional expectation is the expected value of a random variable given specific conditions or information, and for bivariate normal distributions, it can be calculated using a formula that takes into account the means, standard deviations, and correlation coefficient between the two variables. The relationship between conditional expectation and correlation is that they are directly proportional, and conditional expectation can be negative for bivariate normal distributions when the correlation between the variables is negative.
  • #1
Ryuuzakie
3
0
Hey guys, I'm having a bit of a problem with this question...


Homework Statement


If X and Y have a bivariate normal distribution with [tex]m_X=m_y=0[/tex] and [tex]\sigma_X=\sigma_Y=1[/tex], find:

a) E(X|Y=1) and Var(X|Y=1)
b) Pr(X+Y>0.5)


Homework Equations


N/A


The Attempt at a Solution


Apologies, but I don't seem to know where to start on this one.
 
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  • #2
how about the joint denisty function?
 

What is a bivariate normal distribution?

A bivariate normal distribution is a probability distribution that describes the relationship between two continuous random variables. It is a type of multivariate normal distribution, which is commonly used in statistics and probability theory.

What is conditional expectation?

Conditional expectation is a statistical concept that refers to the expected value of a random variable, given certain conditions or information. It is denoted as E(X|Y=y), where X is the random variable and Y=y represents the specific condition or information.

How is conditional expectation calculated for bivariate normal distributions?

The conditional expectation for bivariate normal distributions can be calculated using the formula E(X|Y=y) = μx + ρσx(σy)^-1(y-μy), where μx and μy are the means of the two variables, σx and σy are the standard deviations, and ρ is the correlation coefficient between the two variables.

What is the relationship between conditional expectation and correlation for bivariate normal distributions?

For bivariate normal distributions, the conditional expectation and correlation are closely related. The conditional expectation is directly proportional to the correlation coefficient, which means that as the correlation increases, the conditional expectation also increases.

Can conditional expectation be negative for bivariate normal distributions?

Yes, conditional expectation can be negative for bivariate normal distributions. This can happen when the correlation between the two variables is negative, meaning that as one variable increases, the other variable tends to decrease. In such cases, the conditional expectation will be negative if the given condition or information falls below the mean of the second variable.

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