Subject 1. Basic Definitions

A probability distribution specifies the probabilities of the possible outcomes of a random variable.

If you toss a coin 3 times, the possible outcomes are as follows (where H means heads and T means tails): TTT, TTH, THT, HTT, THH, HTH, HHT, HHH.

In total, there are 8 possible outcomes. Of these:

• Only 1 (TTT) has 0 heads occurring.
• Three (TTH, THT and HTT) have 1 heads occurring.
• Three (THH, HTH and HHT) have 2 heads occurring.
• One (HHH) has 3 heads occurring.

Thus, if x = number of heads in 3 tosses of a coin, then x = 0, 1, 2 or 3.

Now, the respective probabilities are 1/8, 3/8, 3/8 and 1/8, as you have just seen. So:

p(0) = p(0 Heads) = 1/8
p(1) = p(1 Head) = 3/8
p(2) = p(2 Heads) = 3/8
p(3) = p(3 Heads) = 1/8

This is a probability distribution; it records probabilities for each possible outcome of the random variable.

Discrete Probability Distribution

A table, graph or rule that associates a probability P(X=x_i) with each possible value x_i that the discrete random variable X can assume is called a discrete probability distribution. It is a theoretical model for the relative frequency distribution of a population.

A random variable is a quantity whose future outcomes are uncertain. Depending on the characteristics of the random variable, a probability distribution may be either discrete or continuous.

A discrete variable is one that cannot take on all values within the limits of the variable. It can assume only a countable number of possible values. For example, responses to a five-point rating scale can only take on the values 1, 2, 3, 4, and 5. The variable cannot have the value 1.7. The variable "number of correct answers on a 100-point multiple-choice test" is also a discrete variable since it is not possible to get 54.12 problems correct. The number of movies you will see this year, the number of trades a broker will perform next month, and the number of securities in a portfolio are all examples of discrete variables.

A continuous variable is one within the limits of variable ranges for which any value is possible. The number of possible values cannot be counted, and, as you will see later, each individual value has zero probability associated with it. For example, the variable "time to solve an anagram problem" is continuous since it could take 2 minutes or 2.13 minutes, etc., to finish a problem. A variable such as a person's height can take on any value as well. The rate of return on an asset is also a continuous random variable since the exact value of the rate of return depends on the desired number of decimal spaces.

Statistics computed from discrete variables are continuous. The mean on a five-point scale could be 3.117 even though 3.117 is not possible for an individual score.

For any random variable, it is necessary to know two things:

• the list of all possible values that the random variable can take on.
• the probability of each value occurring.

These give a probability distribution. The first item on the list is called the range.

With regard to the range of possible outcomes of a specified random variable:

• Sometimes the possible values of a random variable have both lower and upper bounds. For example, there are three possible values of the number of heads showing face-up on two tosses of a coin: 0, 1, and 2. Therefore, the lower bound is 0 and the upper bound is 2.

• Sometimes the lower bound exists, but the upper bound does not. For example, the lower bound of the price of a stock is 0, since it cannot fall below 0. However, there is no upper bound on the price (at least theoretically).

• Sometimes the upper bound exists, but the lower bound does not. Consider the profit or loss of the seller of a call option. Suppose the buyer pays the seller $2 to buy a call option, which gives the buyer the right to buy a stock at $10 by the end of 2006. The maximum profit the seller can make is $2, but the maximum loss the seller may incur is unlimited since there is no upper bound on the possible values of stock prices.

• In other cases, neither bound is obvious. Consider the profit or loss of a big company. In a good year, profits could be as high as dozens of billions of dollars, losses could be equivalent in a very bad year.