1. Show that if C
is a countable subset of S, then P(X 
C) = 0.Virtual Laboratories > Distributions > 1 [2] 3 4 5 6 7 8 9
As usual, suppose that we have a random experiment with a sample space and a probability measure P. A random variable X taking values in a subset S of Rn is said to have a continuous distribution if
P(X = x) = 0 for each x in S.
The fact that X takes any particular value with probability 0 might seem paradoxical at first, but conceptually it is the same as the fact that an interval of R can have positive length even though it is composed of points each of which has 0 length. Similarly, an region of R2 can have positive area even though it is composed of points (or curves) each of which has area 0.
1. Show that if C
is a countable subset of S, then P(X
C) = 0.
Thus, continuous distributions are in complete contrast with discrete distributions, for which all of the probability mass is concentrated on a discrete set. For a continuous distribution, the probability mass is continuously spread over S. Note also that S itself cannot be countable.
Suppose again that X has a continuous distribution on a subset S of Rn. A real-valued function f defined on S is said to be a probability density function for X if f satisfies the following properties:
0 for
x in S.
S
f(x)dx
= 1.A
f(x)dx
= P(X
A) for A 
S.If n > 1, the integrals in properties (b) and (c) are multiple integrals over subsets of Rn, and
dx = dx1 dx2 ··· dxn where x = (x1, x2, ..., xn).
In fact, technically, f is the density of X relative to n-dimensional measure mn, which we recall is given by
mn(A) = A
1dx for A Rn.
Note that mn(S) must be positive (perhaps infinite). In particular,
However, we recall that except for exposition, the low dimensional cases (n = 1, 2, 3) play no special role in probability. Interesting random experiments almost always involve several random variables (that is, a random vector); seldom do we have a single, isolated random variable. Finally, note that we can always extend f to a density on all of Rn by defining f(x) = 0 for x not in S. This extension sometimes simplifies notation.
Property (c) is particularly important since it implies that the probability distribution of X is completely determined by the density function. Conversely, any function that satisfies properties (a) and (b) is a probability density function, and then property (c) can be used to define a continuous distribution on S.
An element x in S that maximizes the density f is called a mode of the distribution. If there is only one mode, it is sometimes used as a measure of the center of the distribution.
Unlike the discrete case, the density function of a continuous distribution is not unique. Note that the values of f on a finite (or even countable) set of points could be changed to other nonnegative values, and properties (a), (b), and (c) would still hold. The key fact is that only integrals of f are important. Another difference is that f(x) can be larger than 1; indeed, f can be unbounded on S. Keep in mind that f(x) is not a probability; it is a probability density: f(x)dx is approximately the probability that X is in an n-dimensional box at x with side lengths dx1, ..., dxn, if these side lengths are small.
2.
Let f(t) = r exp(-rt) for t > 0, where r
> 0 is a parameter. Show that f is a probability density function.
The distribution defined by the density function in the previous exercise is called the exponential distribution with rate parameter r. This distribution is frequently used to model random times, under certain assumptions. The exponential distribution is studied in detail in the chapter on Poisson Processes.
3.
The lifetime T of a certain device (in 1000 hour units) has the
exponential distribution with parameter 1/2. Find P(T > 2).
4. In the exponential
experiment, set r = 1/2.
Run the simulation 1000 times, updating every 10 runs, and note the apparent
convergence of the empirical density function to the true density function.
5.
In Bertrand's problem, a certain random
angle A has density function f(a) = sin(a), 0 < a
<
/
2.
/
4).
6. In Bertrand's
experiment, select the model with
uniform distance. Run the simulation 200 times, updating every run, and compute
the empirical probability of the event {A <
/ 4}. Compare with the true probability in the previous exercise.
7.
Let gn(t) = exp(-t) tn / n!
for t > 0 where n is a nonnegative integer parameter.
Remarkably, we showed in the last section on discrete distributions, that ft(n) = gn(t) is a density function on the nonnegative integers for each t > 0. The distribution defined by the density gn is the gamma distribution; n + 1 is called the shape parameter. The gamma distribution is studied in detail in the chapter on Poisson Processes.
8.
Suppose that the lifetime of a device T (in 1000 hour units) has the
gamma distribution with n = 2. Find P(T > 3).
9. In the gamma
experiment, set r = 1 and k = 3. Run
the experiment 200 times, updating every run. Compute the empirical probability
of the event {T > 3} and compare with the theoretical probability in the
previous exercise.
10. Suppose that g
is a nonnegative function on S. Let
c = S
g(x)dx.
Show that if c is positive and finite, then f(x) = g(x) / c for x in S defines a probability density function on S.
Note that the graphs of g and f look the same, except of a change of scale on the vertical axis. Thus, the result in the last exercise can be used to construct density functions with desired properties (domain, shape, symmetry, and so on). The constant c is sometimes called the normalizing constant.
11. Let g(x)
= x2(1 - x) for 0 < x < 1.
The distribution defined in the last exercise is an example of a beta distribution.
12. Let g(x)
= 1 / xa for x > 1, where a > 0 is a parameter.
1, show that there is no probability density function
proportional to g.The distribution defined in the last exercise is known as the Pareto distribution with shape parameter a.
13. Let g(x)
= 1 / (1 + x2) for x in R.
. The distribution defined in the last exercise is known as the Cauchy distribution, named after Augustin Cauchy. It is a member of the Student t family of distributions.
14. In the
random variable experiment, select the student t distribution.
Set n = 1 to get the Cauchy distribution. Run the simulation 1000 times, updating
every 10 runs. Note how well the empirical density function fits the true density
function.
15.
Let g(z) = exp(-z2 / 2).
)1/2.
Hint: If c denotes the normalizing constant, express c2
has a double integral and convert to polar coordinates.The distribution defined in the last exercise is the standard normal distribution, perhaps the most important distribution in probability.
16. In the
random variable experiment, select the normal distribution
(the default parameters give the standard normal distribution). Run the simulation 1000 times, updating
every 10 runs. Note how well the empirical density function fits the true density
function.
17. Let
f(x,
y) = x + y for 0 < x < 1, 0 < y <
1.
18. Let g(x,
y) = x + y for 0 < x < y <
1.
19. Let g(x,
y) = x2y for 0 < x < 1, 0 < y
< 1.
20. Let g(x,
y) = x2y for 0 < x < y <
1.
21.
Let g(x, y, z) = x + 2y + 3z
for 0 < x < 1, 0 < y < 1, 0 < z < 1.
The following exercises describe an important class of continuous distributions.
22. Suppose that
S is a subset of Rn with positive, finite
measure mn(S). Show that
P(X
A) = mn(A) / mn(S) for A S
if X has the density function in (a).
A random variable X with the density function in Exercise 14 is said to have the continuous uniform distribution on S. The uniform distribution on a rectangle in the plane plays a fundamental role in Geometric Models.
23.
Suppose that (X,
Y) is uniformly distributed on the square S = (-6, 6)2. Find P(X > 0, Y > 0).
24. In
the bivariate uniform
experiment, select square in the list box. Run the
simulation 100 times, updating every run. Watch the points in the scatter plot. Compute
the empirical probability of the event {X > 0, Y > 0} and compare with the true probability.
25.
Suppose that (X,
Y) is uniformly distributed on the triangle S = {(x, y): -6 < y < x < 6}.
Find P(X > 0, Y > 0)
26. In
the bivariate uniform
experiment, select triangle in the list box. Run the
simulation 100 times, updating every run. Watch the points in the scatter plot. Compute
the empirical probability of the event {X > 0, Y > 0} and compare with the true probability.
27.
Suppose that (X,
Y) is uniformly distributed on the circle S = {(x, y): x2 + y2
< 36}. Find P(X > 0, Y > 0).
28. In
the bivariate uniform
experiment, select circle in the list box. Run the
simulation 100 times, updating every run. Watch the points in the scatter plot. Compute
the empirical probability of the event {X > 0, Y > 0} and compare with the true probability.
29. Suppose that (X,
Y, Z) is uniformly distributed on the cube (0, 1)3. Find P(X
< Y < Z)
30.
The time T (in minutes) required to perform a certain job is uniformly distributed
over the interval (15, 60).
Suppose that X is a random variable taking values in S of Rn, with a continuous distribution that has density function f. The density function of X, of course, is based on the underlying probability measure P on the sample space of the experiment, which we will denote by R. This measure could be a conditional probability measure, conditioned on a given event E (a subset R), with P(E) > 0. The usual notation is
f(x | E), x
S.
Note, however, that except for notation, no new concepts are involved. The function above is a continuous density function. That is, it satisfies properties (a) and (b) while property (c) becomes
A
f(x | E)dx = P(X
A | E) for A S.
All results that hold for densities in general have analogues for conditional densities.
31.
Suppose that B
S with P(X
B) = B
f(x)dx > 0. Show that the conditional density of X
given X
B is
B) = f(x) / P(X
B) for x
B.
B) = 0 for x
Bc. 32.
Suppose that
S is a subset of Rn with positive, finite
measure mn(S) and that B
S with mn(B) > 0. Show that if X is
uniformly distributed on S, then the conditional distribution of X
given X
B is uniform on B.
33.
Suppose that (X, Y) has density function f(x, y)
= x + y for 0 < x < 1, 0 < y < 1. Find
the conditional density of (X, Y) given X < 1/2, Y
< 1/2.
If {x1, x2, ..., xn}
Rn is
a data set from a continuous variable X, then an empirical
density function can be computed by partitioning the data range into
subsets of small size, and then computing the density of points in each subset. Empirical
density functions are studied in more detail in the chapter on Random
Samples.
34. For the cicada
data, BW denotes body weight, BL body
length, and G gender. Construct an empirical density function for each of
the following and display each as a bar graph:
35. For the cicada
data, WL denotes wing length and WW wing
width. Construct an empirical density function for (WL, WW).
Unlike the discrete case, the existence of a density function for a continuous distribution is an assumption that we are making. A random variable can have a continuous distribution on a subset S of Rn but with no density function; the distribution is sometimes said to be degenerate. In this subsection, we explore the common ways in such distributions can occur.
First, suppose that X is a random variable taking values in a subset S of Rn with mn(S) = 0. It is possible for X to have a continuous distribution, but X could not have a density relative to mn. In particular, property (c) in the definition could not hold, since the integral on the left would be 0 for any subset A of S. However, in many cases, X may be defined in terms of continuous random variables on lower dimensional spaces that do have densities.
For example, suppose that U is a random variable with a continuous distribution on subset T of Rk (where k < n), and that X = h(U) for some continuous function h from T into Rn. Any event defined in terms of X can be changed into an event defined in terms of U. The following exercise illustrates this situation
36.
Suppose that U is uniformly distributed on the interval (0, 2).
Let X = cos(U), Y = sin(U).
Another situation occurs when a random vector X in Rn (n > 1) has some components with discrete distributions and others with continuous distributions. Such distributions with mixed components are studied in more detail in the section on mixed distributions; however, the following exercise gives an illustration.
37.
Suppose that X is uniformly distributed on {0, 1, 2}, Y is
uniformly distributed on (0, 2), and that X and Y are independent.
Finally, it is also possible to have a continuous distribution on a subset S of Rn with mn(S) > 0, yet still with no density function. Such distributions are said to be singular, and are rare in applied probability. For an example, however, see Bold Play in the chapter Red and Black.