This post extends the discussion of the bivariate normal distribution started in this post from a companion blog. Practice problems are given in the next post.

Suppose that the continuous random variables and follow a bivariate normal distribution with parameters , , , and . What to make of these five parameters? According to the previous post, we know that

- and are the mean and standard deviation of the marginal distribution of ,
- and are the mean and standard deviation of the marginal distribution of ,
- and finally is the correlation coefficient of and .

So the five parameters of a bivariate normal distribution are the means and standard deviations of the two marginal distributions and the fifth parameter is the correlation coefficient that serves to connect and . If , then and are simply two independent normal distributions.

When calculating probabilities involving a bivariate normal distribution, keep in mind that both marginal distributions are normal. Furthermore, the conditional distribution of one variable given a value of the other is also normal. Much more can be said about the conditional distributions.

The conditional distribution of given is usually denoted by or . In additional to being a normal distribution, it has a mean that is a linear function of and has a variance that is constant (it does not matter what is, the variance is always the same). The linear conditional mean and constant variance are given by the following:

Similarly, the conditional distribution of given is usually denoted by or . In additional to being a normal distribution, it has a mean that is a linear function of and has a variance that is constant. The linear conditional mean and constant variance are given by the following:

The information about the conditional distribution of on is identical to the information about the conditional distribution of on , except for the switching of and . An example is helpful.

*Example 1*

Suppose that the continuous random variables and follow a bivariate normal distribution with parameters , , , and . The first two parameters are the mean and standard deviation of the marginal distribution of . The next two parameters are the mean and standard deviation of the marginal distribution of . The parameter is the correlation coefficient of and . Both marginal distributions are normal.

Let’s focus on the conditional distribution of given . It is normally distributed. Its mean and variance are:

The line is also called the least squares regression line. It gives the mean of the conditional distribution of given . Because and are positively correlated, the least squares line has positive slope. In this case, the larger the , the larger is the mean of . The standard deviation of given is constant across all possible values.

With mean and standard deviation known, we can now compute normal probabilities. Suppose the realized value of is 25. Then the mean of is . The standard deviation, as indicated above, is 4. In fact, for any other , the standard deviation of is also 4. Now calculate the probability . We first calculate it using a normal table found here.

Using a TI84+ calculator, . In contrast, the probability is (using the table found here):

Using a TI84+ calculator, . Note that is for the marginal distribution of . It is not conditioned on any realized value of .

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