The Bivariate Normal Distribution

Java Simulation of the bivariate normal experiment

Definition

Suppose that U and V are independent random variables each, with the standard normal distribution. We will need the following five parameters:

Now let X and Y be new random variables defined by

The joint distribution of (X, Y) is called the bivariate normal distribution with the five parameters given above.

Basic Properties

For the following exercises, use properties of mean, variance, covariance, and the normal distribution.

1. Show that X is normally distributed with

2. Show that Y is normally distributed with

3. Show that

4. Show that X and Y are independent if and only if cor(X, Y) = 0

The Bivariate Normal Experiment

In the simulation of the bivariate normal experiment, the random vector (X, Y) has a bivariate normal distribution with 0 means. The standard deviations and correlation can be varied with scroll bars.

2. In the bivariate normal experiment, change the standard deviations of X and Y with the scroll bars. Watch the change in the shape of the probability density functions. Now change the correlation with the scroll bar and note that the probability density functions do not change.

3. In the bivariate normal experiment, set the standard deviation of X to 1.5 and the standard deviation of Y to 0.5. For each of the following correlations, run the experiment 2000 times with an update frequency of 10. Watch the cloud of points in the (X, Y) scatterplot and note the apparent convergence of the empirical density function to the probability density function.

(a) 0 (b) 0.5 (c) -0.5 (d) 0.7 (e) -0.7 (f) 0.9 (g) - 0.9

Density Function

We will now use the change of variables formula to find the joint probability density function (X, Y).

4. Show that inverse transformation is given by

5. Show that the Jacobian of the transformation in Exercise 4 is

Note that the Jacobian is a constant; this is because the transformation in Exercise 1 is linear.

6. Use Exercises 4 and 5, the independence of U and V, and the change of variables formula to show that the joint density of (X, Y) is

If c is a constant, the set of points {(x, y) in R²: f(x, y) = c}is called a level curve of f (these are points of constant probability density).

7. Show that

1. The level curves of f are ellipses centered at (E(X), E(Y))
2. The axes of these ellipses are parallel to the coordinate axes if and only if cor(X, Y)= 0.

8. In the bivariate normal experiment, set the standard deviation of X to 2 and the standard deviation of Y to 1. For each of the following correlations, run the experiment 2000 times with an update frequency of 10 and watch the cloud of points in the (X, Y) scatterplot in light of Exercise 7.

(a) 0 (b) 0.5 (c) -0.5 (d) 0.7 (e) -0.7 (f) 0.9 (g) - 0.9

The following exercise shows that the bivariate normal distribution is preserved under affine transformations.

9. Define

W = a₁X + b₁Y + c₁
Z = a₂X + b₂Y + c₂

Use the change of variables formula to show that (W, Z) has a bivariate normal distribution. Identify the means, variances, and correlation.

10. Show that the conditional distribution of Y given X = x is normal with mean and variance given by

11. Use the representation of X and Y in terms of the independent standard normal variables U and V to show that

Now give another proof of the result in Exercise 10 (note that X and V are independent).

12. In the bivariate normal experiment, set the standard deviation of X to 1.5, the standard deviation of Y to 0.5, and the correlation to 0.7. Run the experiment n =100 times, updating after each run. Compute the following standard error statistic and compare with the standard deviation of Y.

Compare these to the sample standard deviations of X and Y, respectively.

The following problem is a good exercise in using the change of variables formula and will be useful when we discuss the simulation of normal variables.

13. Recall that U and V are independent random variables each with the standard normal distribution. Define the polar coordinates (R, T) of (U, V) by the equations

U = R cos(T), V = R sin(T) where R > 0 and 0 < T < 2 pi

Show that

1. R² has the exponential distribution with parameter 2.
2. T is uniformly distributed on (0, 2 pi).
3. R and T are independent.

The results of this section have straightforward analogues for the general multivariate normal distribution.

Special Distributions