Mixed Distributions

Definition

In a discrete distribution, all of the probability is concentrated on a countable set of points, each of which has positive probability. By contrast, in a continuous distribution, no point has positive probability, but rather the probability is continuously spread out over a "continuum" set. As you might guess, there are probability distributions that have both discrete and continuous components.

Suppose that a random vector X takes values in a subset S Rⁿ. X is said to have a mixed distribution if S can be decomposed into a union of disjoint subsets D and C of Rⁿ with the following properties:

1. D is countable
2. There exists p in (0, 1) such that P(X in D) = p, P(X in C) = 1 - p.
3. P(X = x) > 0 for x in D, P(X = x) = 0 for x in C

Thus, the distribution of X has probability p concentrated at the points in D and the remaining probability 1 - p continuously spread out over C. The following exercise gives a simple example of how such a distribution might arise.

1. Suppose that an Internet provider charges a flat fee of $10 for the first 5 hours or less of connect time and charges $1 per hour for connect time in excess of 5 hours (on a monthly basis). Suppose that the montly conncect time X of a customer has the exponential distribution with parameter r > 0. Let Y denote the monthly charge for the customer. Show that Y has a mixed distribution with D = {10} and C = {y > 10} in the definition above. Find

1. P(Y = 10)
2. P(Y > 10)

Discrete and Continuous Components

Suppose that X has a mixed distribution with the notation given in the definition above.

2. Show that the conditional distribution of X given X in D is purely discrete and has density function f_D given by

f_D(x) = P(X = x) / p.

3. Show that

4. Show that the conditional distribution of X given X in C is purely continuous.

5. Assume that the the conditional distribution of X given X in C has a density f_C. Show that

6. With the assumption in Exercise 5, show that for an arbitrary subset A of Rⁿ,

Distributions