# PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE - Quantitative Techniques for management

Given any random variable, corresponding to a sample space, it is possible to associate probabilities to each of its possible values. For example, in the toss of 3 coins, assuming that they are unbiased, the probabilities of various values of the random variable X, defined in example 1 above, can be written as:

P(X-0)= 1/8, P(X-1)=3/8, P(X-2)= 3/8 and P(X-3)=1/8

The set of all possible values of the random variable X along with their respective probabilities is termed as Probability Distribution of X. The probability distribution of X, defined in example 1 above, can be written in a tabular form as given below:

X : 0 1 2 3 Total

p(X) : 1/8 3/8 3/8 3/8 1

Note that the total probability is equal to unity. In general, the set of n possible values of a random variable X, i.e., {X1, X2, ...... Xn} along with their respective probabilities p(X1), p(X2), ...... p(Xn), where

is called a probability distribution of X. The expression p(X) is called the probability function of X.

Discrete and Continuous Probability Distributions

If a variable can take on any value between two specified values, it is called a continuous variable; otherwise, it is called a discrete variable. Some examples will clarify the difference between discrete and continuous variables.

• Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds.
• Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.

Just like variables, probability distributions can be classified as discrete or continuous. Like any other variable, a random variable X can be discrete or continuous. If X can take only finite or countably infinite set of values, it is termed as a discrete random variable. On the other hand, if X can take an uncountable set of infinite values, it is called a continuous random variable.

Discrete Probability Distributions

If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution. With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution can always be presented in tabular form.

Continuous Probability Distributions

If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution. The equation used to describe a continuous probability distribution is called a probability density function.

Sometimes, it is referred to as a density function (PDF) The random variable defined in example 1 is a discrete random variable. However, if X denotes the measurement of heights of persons or the time interval of arrival of a specified number of calls at a telephone desk, etc., it would be termed as a continuous random variable.

The distribution of a discrete random variable is called the Discrete Probability Distribution and the corresponding probability function p(X) is called a Probability Mass Function. In order that any discrete function p(X) may serve as probability function of a discrete random variable X, the following conditions must be satisfied :
(i) p(Xi)≥ 0 ∀i = 1, 2, ...... n and

(ii)

The density function has the following properties:

• Since the continuous random variable is defined over a continuous range of values (called the domain of the variable), the graph of the density function will also be continuous over that range.
• The area bounded by the curve of the density function and the x-axis is equal to 1, when computed over the domain of the variable.
• The probability that a random variable assumes a value between a and b is equal to the area under the density function bounded by a and b.

The conditions for any function of a continuous variable to serve as a probability density function are :

(i) p(X)≥ 0 real values of X, and

Remarks:

1. When X is a continuous random variable, there are an infinite number of points in the sample space and thus, the probability that X takes a particular value is always defined to be zero even though the event is not regarded as impossible. Hence, we always talk of the probability of a continuous random variable lying in an interval.
2. The concept of a probability distribution is not new. In fact it is another way of representing a frequency distribution. Using statistical definition, we can treat the relative frequencies of various values of the random variable as the probabilities.

Example : Two unbiased die are thrown. Let the random variable X denote the sum of points obtained. Construct the probability distribution of X.

Solution: The possible values of the random variable are :

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

The probabilities of various values of X are shown in the following table:

Probability Distribution of X

Example : Three marbles are drawn at random from a bag containing 4 red and 2 white marbles. If the random variable X denotes the number of red marbles drawn, construct the probability distribution of X.

Solution: The given random variable can take 3 possible values, i.e., 1, 2 and 3. Thus, we can compute the probabilities of various values of the random variable as given below:

P(X = 1, i.e., 1R and 2 W marbles are drawn) = 4C1 x 2C2 / 6C3 = 4/20
P(X = 2, i.e., 2R and 1W marbles are drawn) = 4C2 x 2C1 / 6C3 = 12/20
P(X = 3, i.e., 3R marbles are drawn) = 4C3 / 6C3 = 4/20

Note: In the event of white balls being greater than 2, the possible values of the random variable would have been 0, 1, 2 and 3.
A continuous probability distribution differs from a discrete probability distribution in several ways.

• The probability that a continuous random variable will assume a particular value is zero.
• As a result, a continuous probability distribution cannot be expressed in tabular form.
• Instead, an equation or formula is used to describe a continuous probability distribution.

Cumulative Probability Function or Distribution Function

This concept is similar to the concept of cumulative frequency. The distribution function is denoted by F(x).

For a discrete random variable X, the distribution function or the cumulative probability function is given by F(x) = P(X ≤x).
If X is a random variable that can take values, say 0, 1, 2, ......, then

F(1) = P(X = 0) + P(X =1), F(2) = P(X = 0) + P(X =1) +P(X = 2), etc.

Similarly, if X is a continuous random variable, the distribution function or cumulative probability density function is given by