1. Show that X
is normally distributed with mean µ1 and standard deviation d1.Virtual Laboratories > Special Distributions > 1 2 3 4 5 6 [7] 8 9 10 11 12 13 14 15
Suppose that U and V are independent random variables each, with the standard normal distribution. We will need the following five parameters:
µ1 and µ2 in R, d1 and d2 > 0, and p in [-1, 1].
Now let X and Y be new random variables defined by
The joint distribution of (X, Y) is called the bivariate normal distribution with parameters µ1, µ2, d1, d2 and p.
For the following exercises, use properties of mean, variance, covariance, and the normal distribution.
1. Show that X
is normally distributed with mean µ1 and standard deviation d1.
2. Show that Y
is normally distributed with with mean µ2 and standard deviation d2.
3. Show that cor(X,
Y) = p.
4. Show that X
and Y are independent if and only if cor(X, Y) = 0.
5. 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.
6. 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: p = 0, p = 0.5,
p = -0.5, p = 0.7, p = -0.7, p = 0.9,
p = -0.9.
We will now use the change of variables formula to find the joint probability density function (X, Y).
7. Show
that inverse transformation is given by
8. Show
that the Jacobian of the transformation in the previous exercise is
d(u, v) / d(x, y) = 1 / [d1d2(1 - p2)1/2].
Note that the Jacobian is a constant; this is because the transformation is linear.
9. Use the
previous exercises, the independence of U and V, and the change of variables
formula to show that the joint density of (X, Y) is
f(x, y) = C exp[Q(x, y)]
where the normalizing constant C and the quadratic form Q are given by
d1d2(1
- p2)1/2]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).
10. Show that
11. 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: p = 0, p = 0.5, p = -0.5,
p = 0.7, p = -0.7, p = 0.9, p = -0.9.
The following exercise shows that the bivariate normal distribution is preserved under affine transformations.
12. Define
W = a1X + b1Y + c1
and
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.
13. Show that the
conditional distribution of Y given X = x is normal with mean and variance
given by
14. Use the
representation of X and Y in terms of the independent standard normal
variables U and V to show that
Y = µ2 + d2 p (X - µ1) / d1 + d2 (1 - p2)1 / 2 V.
Now give another proof of the result in Exercise 13 (note that X and V are independent).
15. 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.
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.
16. 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.
Show that
).The results of this section have straightforward analogues for the general multivariate normal distribution.