Discrete Random Variables

A probability distribution is a table of values showing the probabilities of various outcomes of an experiment.

For example, if a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3. The probabilities of each of these possibilities can be tabulated as shown:

A discrete variable is a variable which can only take a countable number of values. In this example, the number of heads can only take 4 values (0, 1, 2, 3) and so the variable is discrete. The variable is said to be random if the sum of the probabilities is one. 

Probability Density Function

The probability density function (p.d.f.) of X (or probability mass function) is a function which allocates probabilities. Put simply, it is a function which tells you the probability of certain events occurring. The usual notation that is used is P(X = x) = something. The random variable (r.v.) X is the event that we are considering. So in the above example, X represents the number of heads that we throw. So P(X = 0) means "the probability that no heads are thrown". Here, P(X = 0) = 1/8 (the probability that we throw no heads is 1/8 ).

In the above example, we could therefore have written:

Quite often, the probability density function will be given to you in terms of x. In the above example, P(X = x) = ³C_x / (2)³  (see permutations and combinations for the meaning of ³C_x).

Example

A die is thrown repeatedly until a 6 is obtained. Find the probability density function for the number times we throw the die.

Let X be the random variable representing the number of times we throw the die.

P(X = 1) = 1/6 (if we only throw the die once, we get a 6 on our first throw. The probability of this is 1/6 ).

P(X = 2) = (5/6) × (1/6) (if we throw the die twice before getting a 6, we must throw something that isn't a 6 with our first throw, the probability of which is 5/6 and we must throw a 6 on our second throw, the probability of which is 1/6)

etc

In general, P(X = x) = (5/6)^(x-1) × (1/6)

Cumulative Distribution Function

The cumulative distribution function (c.d.f.) of a discrete random variable X is the function F(t) which tells you the probability that X is less than or equal to t. So if X has p.d.f. P(X = x), we have:

F(t) = P(X £ t) = SP(X = x).

In other words, for each value that X can be which is less than or equal to t, work out the probability that X is that value and add up all such results.

Example

In the above example where the die is thrown repeatedly, lets work out P(X £ t) for some values of t.

P(X £ 1) is the probability that the number of throws until we get a 6 is less than or equal to 1. So it is either 0 or 1. 

P(X = 0) = 0 and P(X = 1) = 1/6. Hence P(X £ 1) = 1/6

Similarly, P(X £ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0 + 1/6 + 5/36 = 11/36