Random Variables

For many of the experiments that we model, we are not interested in the outcomes themselves, but instead in some numerical attribute associated with the outcome. In the example above that modelled the games played by two gamblers, we were not interested in the particular outcome, $WWWWLLLLLL$, but instead all that was important was the number of wins, in this case 4, associated with the outcome. This numerical attribute, the number of wins, is an attribute possessed by each possible outcome of the experiment. We call such numerical attributes random variables.

Formally, a random variable is a function, $X:\Omega\rightarrow\Re$, that assigns a real number to each outcome of an experiment. In the case of the gambler's problem, if $X$ denotes the number of wins in 10 games, and if $\omega=\{WWWWLLLLLL\}$, then $X(\omega)=4$. If all that we care to observe in an experiment is the value of $X$, then the only events we need to work with are events defined in terms of the random variable. For example,

$$\{\omega\in\Omega:X(\omega)\le 5\}$$

is an event defined in terms of $X$, and its probability,

$$P(\omega\in\Omega:X(\omega)\le 5)$$

is ordinarily expressed as $P(X\le 5)$. Note that this notation suppresses the fact that $X$ is really a function, not a number.

One requirement that we have regarding random variables is that we must be able to obtain the probability of events in which the random variable belongs to an interval of real numbers, along with all the sets one can obtain by finite unions, intersections, and complements of such events. The probabilities of events of this form can be obtained from the distribution function (d.f.) of the random variable, defined by

$$F(x) = P(X\le x),\ -\infty < x < \infty.$$

For example,

$$\begin{eqnarray*} P(a<X\le b) &=& P(\{X\le b\} \bigcap \{X\le a\}^c) \\ &=& P(X\le b) - P(X\le a) \\ &=& F(b) - F(a),\ -\infty < a<b < \infty. \end{eqnarray*}$$

Another example:

$$P(X>a) = P(\{X\le a\}^c) = 1 - P(X\le a) = 1 - F(a).$$

Note: we make a notational distinction between the random variable and a value of the random variable by using upper case letters to denote random variables and lower case letters to denote values of a random variable.

The sample space of a random variable is the set of all possible values of the random variable. If the sample space is finite or countably infinite, we say that the random variable is discrete. If the distribution function is differentiable, we say that the random variable is continuous. If $X$ is a discrete random variable, it is usually easier to obtain probabilities of events in terms of its probability mass function (pmf), defined by $p(x) = P(X=x)$. The correspondence between the d.f. of a random variable and its pmf is given by the following relationships:

$$F(x) = \sum_{a\le x} p(a),\ \ p(a) = F(a) - F(a-),$$

where $F(a-)$ denotes the limit from below of the d.f. This implies that the d.f. of a discrete random variable is a step-function. In the case of an integer-valued random variable, we have,

$$F(n) = \sum_{i\le n}p(i),\ \ p(n) = F(n) - F(n-1).$$

Likewise,

$$P(X\in A) = \sum_{n\in A} p(n).$$

The basic axioms of probability can be used to show that distribution functions of random variables satisfy the following properties:

1. $F(x)$ is a monotone non-decreasing function.
2. 
   
   $$\lim_{x\rightarrow -\infty} F(x) = 0.$$
   
   

3. 
   
   $$\lim_{x\rightarrow \infty} F(x) = 1.$$
   
   

Any function that satisfies these properties is the distribution function of some random variable.

Probability mass functions satisfy the properties,

1. $p(x)>0$ for only a finite or countably infinite number of values of $x$, and is 0 for all other values of $x$.
2. 
   
   $$\sum_x p(x) = 1.$$
   
   

A function that satisfies these properties is the p.m.f. of some random variable.

In the previous section, we obtained the probability of randomly selecting $k$ females for a subcommittee of size 5 from a committee that has 40 males and 20 females. Let $N$ denote the number of females selected for the subcommittee. Then the p.m.f. of this random variable is given by,

$$p(k) = P(N=k) = \frac{ {20\choose k} {40\choose {5-k}} }{ {60\choose 5} }, \ \ 0\le k\le 5.$$

Note that $p(x)=0$ for $x$ not equal to one of the values, $0,1,2,3,4,5$.