1. Show that the
chi-square distribution with n degrees of freedom has density functionVirtual Laboratories > Special Distributions > 1 2 3 [4] 5 6 7 8 9 10 11 12 13 14 15
In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of the sample variance when the underlying distribution is normal and in a goodness of fit test.
For n > 0, the gamma distribution with shape parameter k = n / 2 and scale parameter 2 is called the chi-square distribution with n degree of freedom.
1. Show that the
chi-square distribution with n degrees of freedom has density function
f(x) = xn/2 - 1exp(-x / 2) / [2n/2 gam(n / 2)] for x > 0.
2. In the
random variable
experiment, select the chi-square distribution. Vary n
with the scroll bar and note the shape of the density function. With n = 5, run
the simulation 1000 times with an update frequency of 10 and note the apparent convergence
of the empirical density function to the true density function.
3. Show that the
chi-square distribution with 2 degrees of freedom is the exponential distribution with
scale parameter 2.
4. Draw a careful
sketch of the chi-square density functions in each of the following cases:
The distribution function and the quantile function do not have simple, closed-form representations. Approximate values of these functions can be obtained from the table of the chi-square distribution and from the quantile applet.
5. In the quantile
applet, select the chi-square distribution. Vary the
parameter and note the shape of the density function and the distribution
function. In each of the following cases, find the median, the first and third
quartiles, and the interquartile range.
The mean, variance, moments, and moment generating function of the chi-square distribution can be obtained easily from general results for the gamma distribution. In the following exercises, suppose that X has the chi-square distribution with n degrees of freedom.
6. Show
that
7.
Show that
E(Xk) = 2k gam(n/2 + k) / gam(n/2).
8. Show
that
E[exp(tX)] = (1 - 2t)-n/2 for t < 1/2.
9. In the
simulation of the random variable
experiment, select the chi-square distribution. Vary n
with the scroll bar and note the size and location of the mean/standard deviation bar.
With n = 4, run the simulation 1000 times with an update frequency of 10 and note
the apparent convergence of the empirical moments to the distribution moments.
10. Suppose
that Z has the standard normal distribution. Use change
of variable techniques to show that U = Z2 has the chi-square
distribution with 1 degree of freedom.
11. Use moment
generating functions or properties of the gamma distribution to show that if X has
the chi-square distribution with m degrees of freedom, Y has the
chi-square distribution with n degrees of freedom, and X and Y are
independent, then X + Y has the chi-square distribution with m + n
degrees of freedom.
12. Suppose
that Z1, Z2, ..., Zn are
independent standard normal variables (that is, a
random sample of size n from the standard
normal distribution). Use the results of the previous two exercises to show that
V = Z12 + Z22 + ··· + Zn2
has the chi-square distribution with n degrees of freedom.
The result of the last exercise is the reason that the chi-square distribution deserves a name of its own. Sums of squares of independent normal variables occur frequently in statistics. On the other hand, the following exercise shows that any gamma distributed variable can be rescaled into a variable with a chi-square distribution.
13. Suppose that X
has the gamma distribution with shape parameter k and scale parameter b.
Show that Y = 2X / b has the chi-square distribution
with 2k degrees of freedom.
14. Suppose that a
missile is fired at a target at the origin of a plane coordinate system, with units in
meters. The missile lands at (X, Y) where X and Y
are independent and each has the normal distribution with mean 0 and variance 100. The
missile will destroy the target if it lands within 20 meters of the target. Find the
probability of this event.
From the central limit theorem, and previous results for the gamma distribution, it follows that if n is large, the chi-square distribution with n degrees of freedom can be approximated by the normal distribution with mean n and variance 2n. More precisely, if X has the chi-square distribution with n degrees of freedom, then the distribution of the standardized variable
(X - n) / (2n)1/2,
converges to the standard normal distribution as n increases to infinity:
15. In the
simulation of the random variable
experiment, select the chi-square distribution. Start
with n = 1 and increase n. Note the shape of the density function. With n
= 20, run the experiment 1000 times with an update frequency of 10 and note the apparent
convergence of the empirical density function to the true density function.
16. Suppose that X
has the chi-square distribution with n = 18 degrees of freedom. For
each of the following, compute the true value using the quantile
applet and then compute the normal approximation. Compare.