2.3 random variables and probability distributions 7

The product operator is defined as

Again, where there is no confusion as to the limits of /, we will just write fl, X, or W X,. As with the 2 operator, we can also use the double-product operator, - For instance, if we have variables X,, X2, X- and „ Y2, then llIlKYj = X,X2X,Y,Y2.

2.3 Random Variables and Probability Distributions

A variable X is said to be a random variable (rv) if for every real number a there exists a probability P(X < a) that X takes on a value less than or equal to a. We shall denote random variables by capital letters X, Y, Z, and so on. We shall use lowercase letters, x, y, z, and so on, to denote particular values of the random variables. Thus P{X = x) is the probability that the random variable X takes the value x. P{xi < Z < 3) is the probabiUty that the random variable X takes values between :, and 2, both inclusive.

If the random variable X can assume only a particular finite (or countably infinite) set of values, it is said to be a discrete random variable. A random variable is said to be continuous if it can assume any value in a certain range. An example of a discrete random variable is the number of customers arriving at a store during a certain period (say, the first hour of business). An example of a continuous random variable is the income of a family in the United States. In actual practice, use of continuous random variables is popular because the mathematical theory is simpler. For instance, when we say that income is a continuing random variable, we do not mean that it is continuous (in fact, strictly speaking, it is discrete) but that it is a convenient approximation to treat it that way.

A formula giving the probabilities for different values of the random variable X is called a probability distribution in the case of discrete random variables, and probability density function (denoted by p.d.f.) for continuous random variables. This is usually denoted by/(x).

In general, for a continuous random variable, the occurrence of any exact value of X may be regarded as having a zero probability. Hence probabilities are discussed in terms of some ranges. These probabilities are obtained by integrating fix) over the desired range. For instance, if we want Prob(a s Z < b), this is given by

Joint, Marginal, and Conditional Distributions

We are often interested in not just one random variable but in the relationship between several random variables. Suppose that we have two random variables, X and Y. Now we have to consider:

1. The joint p.d.f.:/(A:, j).

2. The marginal p.d.f.s:/( ) and/ ).

3. The conditional p.d.f.s:

(a) /( : ), which is the distribution of X given = y.

(b) / :), which is the distribution of given that X = x.

The joint density can be written as the product of the marginal and conditional densities. Thus

fix, y) = f{x)f{y\x)

= fiy)Ax\y)

If fix, y) = fix)fiy) for all X and y, then x and are said to be independent. Note that if they are independent,

f(x\y) = fix) and fiy\x) = fiy)

that is, the conditional distributions are the same as the marginals. This makes intuitive sense because for X, whether or not Y is fixed at a certain level, is irrelevant. Similarly, for F it should be irrelevant at what level we fix X.

Illustrative Example

Consider, for instance, the discrete distribution of X and Y defined by the following probabilities:

fiy) f{y\x = 3) fiy\x = 4) fiy\x = 5)

The probability that the random variable X takes on values at or below a number is often written as f (c) = ProbCX s c). The function F{x) represents, for different values of x, the cumulated probabilities and hence is called the cumulative disstribution function (denoted by c.d.f.) Thus

2.4 the normal probabiuty distribution and related distributions 19

Since the conditional distributions of depend on the values of x, X and Y cannot be independent. On the other hand, if the distribution of X and Y is defined as

/(y) = \ = 3) = f{y\x = 4) = f{y\x = 5) 0.5 0.5 0.5 0.5

0.3 0.3 0.3 0.3

0.2 0.2 0.3 0.2

we see that the conditional distributions of for the different values of x and the marginal distribution of are the same and hence X and Fare independent.

2 The Normal Probability Distribution and Related Distributions

If we are given the probability distribution of a random variable X, we can determine the probability that X lies in an interval (a, b). There are some probability distributions for which the probabilities have been tabulated and which are considered suitable descriptions for a wide variety of phenomena. These are the normal distribution and the /, x and F distributions. We discuss these and also the lognormal and bivariate normal distributions. There are other distributions as well, such as the gamma and beta distributions, for which extensive tables are available. In fact, the x-distribution is a particular case of the gamma distribution and the t and F distributions are particular cases of the beta distribution. We do not need all these relationships here.

There is also a question of whether the normal distribution is an appropriate one to use to describe economic variables. However, even if the variables are not normally distributed, one can consider transformations of the variables so that the transformed variables are normally distributed.

The Normal Distribution

The normal distribution is a bell-shaped distribution which is used most extensively in statistical applications in a wide variety of fields. Its probability density function is given by

-00 < - < 4- 00

Its mean is p. and its variance is or. When x has the normal distribution with mean jx and variance , we write this compactly as : ~ N(jl, or).