$$\renewcommand{\P}{\mathbb{P}}$$ $$\newcommand{\R}{\mathbb{R}}$$

## 3. Mixed Distributions

### Basic Theory

As usual, we start with a random experiment with probability measure $$\P$$ 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 $$S \subseteq \R^n$$. Then $$X$$ has a distribution of mixed type if $$S$$ can be partitioned into subsets $$D$$ and $$C$$ with the following properties:

1. $$D$$ is countable and $$0 \lt \P(X \in D) \lt 1$$.
2. $$\P(X = x) = 0$$ for all $$x \in C$$.

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 $$p = \P(X \in D)$$, so that $$0 \lt p \lt 1$$. We can define a function on $$D$$ that is a partial probability density function for the discrete part of the distribution.

Let $$g(x) = \P(X = x)$$ for $$x \in D$$. Then

1. $$g(x) \ge 0$$ for $$x \in D$$
2. $$\sum_{x \in D} g(x) = p$$
3. $$\P(X \in A) = \sum_{x \in A} g(x)$$ for $$A \subseteq D$$

Usually, the continuous part of the distribution is also described by a partial probability density function. Thus, suppose there is a nonnegative function $$h$$ on $$C$$ such that

$\P(X \in A) = \int_A h(x) dx, \quad A \subseteq C$

$$\int_C h(x) dx = 1 - p$$.

The distribution of $$X$$ is completely determined by the partial probability density functions $$g$$ and $$h$$. First, we extend the functions $$g$$ and $$h$$ to in the usual way: $$g(x) = 0$$ for $$x \in C$$, and $$h(x) = 0$$ for $$x \in D$$.

For $$A \subseteq S$$,

$\P(X \in A) = \sum_{x \in A} g(x) + \int_A h(x) dx$

The conditional distributions on $$D$$ and on $$C$$ are purely discrete and continuous, respectively.

The conditional distribution of $$X$$ given $$X \in D$$ is discrete, with probability density function

$f(x | X \in D) = \frac{g(x)}{p}, \quad x \in D$

The conditional distribution of $$X$$ given $$X \in C$$ is continuous, with probability density function

$f(x | X \in C) = \frac{h(x)}{1 - p}, \quad x \in C$

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 $$T \in [0, \infty)$$ 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 $$T$$ at $$a$$, as follows:

$U = \begin{cases} T, & T \lt a \\ a, & T \ge a \end{cases}$

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

1. $$D = \{a\}$$ and $$g(a) = \int_a^\infty f(t) dt$$
2. $$C = [0, a)$$ and $$h(t)= f(x)$$ for $$x \in [0, a)$$

Suppose that random variable $$X$$ has a continuous distribution on $$\R$$, with probability density function $$f$$. The variable is truncated at $$a$$ and $$b$$ ($$a \lt b$$) to create a new random variable $$Y$$ as follows:

$Y = \begin{cases} a, & X \le a \\ X, & a \lt X \lt b \\ b, & X \ge b \end{cases}$

$$Y$$ has a mixed distribution. In particular

1. $$D = \{a, b\}$$, $$g(a) = \int_{-\infty}^a f(x) dx$$, $$g(b) = \int_b^\infty f(x) dx$$
2. $$C = (a, b)$$ and $$h(x) = f(x)$$ for $$x \in (a, b)$$

#### 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 $$T \subseteq \R^n$$.

$$\P[(X, Y) = (x, y)] = 0$$ for $$(x, y) \in S \times T$$. Thus $$(X, Y)$$ has a continuous distribution on $$S \times T$$.

Usually, $$(X, Y)$$ has a probability density function $$f$$ on $$S \times T$$ in the following sense:

$\P[(X, Y) \in A \times B] = \sum_{x \in A} \int_B f(x, y) dy, \quad A \times B \subseteq S \times T$

More generally, for $$C \subseteq S \times T$$ and $$x \in S$$, define the cross section of $$C$$ at $$x$$ by $$C(x) = \{y \in T: (x, y) \in C\}$$. Then

$\P[(X, Y) \in C] = \sum_{x \in S} \int_{C(x)} f(x, y) dy, \quad C \subseteq S \times T$

Technically, $$f$$ is the probability density function of $$(X, Y)$$ with respect to the product measure on $$S \times T$$ formed from counting measure $$\#$$ on $$S$$ and $$n$$-dimensional measure $$\lambda_n$$ on $$T$$.

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 $$\frac{1}{2}$$ uniformly distributed on the set $$\{1, 2, \ldots, 8\}$$ and has probability $$\frac{1}{2}$$ uniformly distributed on the interval $$[0, 10]$$. Find $$\P(X \gt 6)$$.

$$\frac{13}{40}$$

Suppose that $$(X, Y)$$ has probability $$\frac{1}{3}$$ uniformly distributed on $$\{0, 1, 2\}^2$$ and has probability $$\frac{2}{3}$$ uniformly distributed on $$[0, 2]^2$$. Find $$\P(Y \gt X)$$.

$$\frac{4}{9}$$

Suppose that the lifetime $$T$$ of a device (in 1000 hour units) has the exponential distribution with probability density function $$f(t) = e^{-t}$$ for $$0 \le t \lt \infty$$. A test of the device is terminated after 2000 hours; the truncated lifetime $$U$$ is recorded. Find each of the following:

1. $$\P(U \lt 1)$$
2. $$\P(U = 2)$$
1. $$1 - e^{-1} \approx 0.6321$$
2. $$e^{-2} \approx 0.1353$$

Let

$f(x, y) = \begin{cases} \frac{1}{3}, & x = 1, \, 0 \le y \le 1 \\ \frac{1}{6}, & x = 2, \, 0 \le y \le 2 \\ \frac{1}{9}, & x = 3, \, 0 \le y \le 3 \end{cases}$
1. Show that $$f$$ is a mixed probability density function in the sense defined above, with $$S = \{1, 2, 3\}$$ and $$T = [0, 3]$$.
2. Find $$\P(X \gt 1, Y \lt 1)$$.
1. $$\frac{5}{18}$$
Let $$f(p, k) = 6 \binom{3}{k} p^{k + 1} (1 - p)^{4 - k}$$ for $$k \in \{0, 1, 2, 3\}$$ and $$p \in [0, 1]$$.
1. Show that $$f$$ is a mixed probability density function in the sense defined above.
2. Find $$\P(V \lt \frac{1}{2}, X = 2)$$ where $$(X, V)$$ is a random vector with probability density function $$f$$.
1. $$\frac{33}{320} \approx 0.1031$$
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 $$(N, W)$$.