Mixed Distributions

As usual, we start with a random experiment with probability measure

on an underlying sample space. In this section, we will discuss two mixed cases for the distribution of a random variable: the case where the distribution is partly discrete and partly continuous, and the case where the variable has both discrete coordinates and continuous coordinates.

Distributions of Mixed Type

Suppose that

X

is a random variable for the experiment, taking values in

S n

. Then

X

has a distribution of mixed type if

S

can be partitioned into subsets

D

and

C

with the following properties:

Thus, part of the distribution of

X

is concentrated at points in a discrete set

D

; the rest of the distribution is continuously spread over

C

. In the picture below, the light blue shading is intended to represent a continuous distribution of probability while the darker blue dots are intended to represents points of positive probability.

Let

p X D

, so that

0 p 1

. We can define a function on

D

that is a partial probability density function for the discrete part of the distribution.

Let $g x X x$ for $x D$ . Show that

$g x 0$ for $x D$
$x D g x p$
$X A x A g x$ for $A D$

Usually, the continuous part of the distribution is also described by a partial probability density function. Thus, suppose there is a nonnegative function

h

C

such that

Show that $x C h x 1 p$ .

The distribution of

X

is completely determined by the partial probability density functions

g

and

h

. First, we extend the functions

g

and

h

S

in the usual way:

g x 0

for

x C

, and

h x 0

for

x D

Show that

X A x A g x x A h x, A S

The conditional distributions on

D

and on

C

are purely discrete and continuous, respectively.

Show that the conditional distribution of $X$ given $X D$ is discrete, with probability density function

f x X D g x p, x D

Show that the conditional distribution of $X$ given $X C$ is continuous, with probability density function

f x X C h x 1 p, x C

Thus, the distribution of

X

is a mixture of a discrete distribution and a continuous distribution. Mixtures are studied in more generality in the section on conditional distributions.

Truncated Variables

Distributions of mixed type occur naturally when a random variable with a continuous distribution is truncated in a certain way. For example, suppose that

T 0

is the random lifetime of a device, and has a continuous distribution with probability density function

f

. In a test of the device, we can't wait forever, so we might select a positive constant

a

and record the random variable

U

, defined by truncating

T

a

as follows:

Show that $U$ has a mixed distribution. In particular, show that, in the notation above,

$D a$ and $g a t a f t$
$C 0 a$ and $h t f t$ for $t 0 a$

Suppose that random variable

X

has a continuous distribution on

, with probability density function

f

. The variable is truncated at

a

and

b

(

a b

) to create a new random variable

Y

as follows:

Show that $Y$ has a mixed distribution. In particular show that

$D a b$ , $g a x a f x$ , $g b x b f x$
$C a b$ and $h x f x$ for $x a b$

Random Variable with Mixed Coordinates

Suppose

X

and

Y

are random variables for our experiment, and that

X

has a discrete distribution, taking values in a countable set

S

while

Y

has a continuous distribution on

T n

Show that $X Y x y 0$ for $x y S T$ . Thus $X Y$ has a continuous distribution on $S T$ .

Usually,

X Y

has a probability density function

f

S T

in the following sense:

More generally, for $C S T$ and $x S$ , define the cross section at $x$ by $C x y T x y C$ Show that

X Y C x S y C x f x y, C S T

Technically,

f

is the probability density function of

X Y

with respect to the product measure on

S T

formed from counting measure on

S

and

n

-dimensional measure

n

T

Random vectors with mixed coordinates arise naturally in applied problems. For example, the cicada data set has 4 continuous variables and 2 discrete variables. The M&M data set has 6 discrete variables and 1 continuous variable. Vectors with mixed coordinates also occur when a discrete parameter for a continuous distribution is randomized, or when a continuous parameter for a discrete distribution is randomized.

Examples and Applications

Suppose that $X$ has probability $1 2$ uniformly distributed on the set $12 8$ and has probability $1 2$ uniformly distributed on the interval $010$ . Find $X 6$ .

Suppose that $X Y$ has probability $1 3$ uniformly distributed on $0122$ and has probability $2 3$ uniformly distributed on $022$ Find $Y X$ .

Suppose that the lifetime $T$ of a device (in 1000 hour units) has the exponential distribution with probability density function $f t t, t 0$ . A test of the device is terminated after 2000 hours; the truncated lifetime $U$ is recorded. Find each of the following:

$U 1$
$U 2$

Let

f x y 1 3 x 1, 0 y 1 1 6 x 2, 0 y 2 1 9 x 3, 0 y 3

Show that $f$ is a mixed density in the sense defined above, with $S 123$ and $T 03$
Find $X 1 Y 1$ .

Let $f p k 6 3 k p k 1 1 p 4 k$ for $k 0123$ and $p 01$ .

Show that $f$ is a mixed probability density function in the sense defined above.
Find $V 1 2 X 2$ where $V X$ is a random vector with probability density function $f$ .

As we will see in the section on conditional distributions, the distribution in the last exercise models the following experiment: a random probability

V

is selected, and then a coin with this probability of heads is tossed 3 times;

X

is the number of heads.

For the M&M data, let $N$ denote the total number of candies and $W$ the net weight (in grams). Construct an empirical density function for $N W$

3. Mixed Distributions

Basic Theory

Distributions of Mixed Type

Truncated Variables

Random Variable with Mixed Coordinates

Examples and Applications