Mixed Distributions

Mixed Distributions

We have restricted our discussion entirely to random variables which are either
discrete or continuous. Such random variables are certainly the most important
in applications. However, there are situations in which we may encounter the
mixed type: the random variable X may assume certain distinct values, say
x₁. . . . , x_n, with positive probability and also assume all values in some interval,
say a ≤ x ≤ b. The probability distribution of such a random variable would
be obtained by combining the ideas considered above for the description of discrete
and continuous random variables as follows. To each value x; assign a number
p(x_i) such that p(x_i) ≥ 0, all i, and such that
\[\sum\limits_{i=1}^{n}{p({{x}_{i}})\text{ }=\text{ }p\text{ }<\text{ }1}\]

 Then define a function f satisfying

\[\begin{align}

  & f\left( x \right)\ge 0,\int_{a}^{b}{f\left( x \right)}dx=1-p \\

 &  \\

\end{align}\]
 For all a, b, with -∞

 \[p\left( a\le X\le b \right)=\int_{a}^{b}{f(x)dx}+\sum\limits_{[i:a\le {{x}_{i}}\le b]}{p\left( {{x}_{i}} \right)}\]

In this way we satisfy the condition

\[P\left( S \right)=P\left( -\infty

A random variable of mixed type might arise as follows. Suppose that we are
testing some equipment and we let X be the time of functioning. In most problems
we would describe X as a continuous random variable with possible values x ≥ 0.
However, situations may arise in which there is a positive probability that the item
does not function at all, that is, it fails at time X = 0. In such a case we would
want to modify our model and assign a positive probability, say p, to the outcome
X = 0. Hence we would have P(X = 0) = p and P(X > 0) = 1 - p. Thus
the number p would describe the distribution of X at 0, while the pdf f would
describe the distribution for values of X > 0