Introduction to Discrete Probability Distributions

Probability distributions are concerned with the probability of a random variable equalling a certain value, or range of values. The random variable can take any value in a specified range, and each possible value has an associated probability.

The discrete bit means that the random variable (often called X) can only take a finite number of specific values (e.g. 1,2,3,4 etc.). The opposite is continuous, which means the variable can take up any value over a range, and is only restricted by the accuracy of measurement (e.g. height).

A probability density function is a list of all the possible outcomes and their associated probabilities. These can be shown as a table or a formula e.g.

The probability density function "number of 6s when 2 dice are rolled" can be written as;

x	p(X=x)
0
1
2

Or alternatively :

  Note that P(X=x) is the same as p(x). It means "the probability that a random variable will take the value x.

Note that the sum of the probabilities must be 1.

Also note that each individual probability must lie between 0 and 1.

 

Expectation and Variance

The expectation of a random variable is the expected result. It is the equivalent of the mean of a frequency distribution.

For a frequency distribution, the mean is thus the expectation of a d.p.d. is

However, since the sum of the probabilities will always equal 1, we can say, [in formula book].

So, the expectation of X from above can be worked out;

x	P(x)	xP(x)
0		0
1
2

So, from this we conclude that, on average, the number of 6s expected when two die are thrown is 1/3.

The formula for variance can also be easily understood by referring to the formula used for frequency

distributions, i.e. This means "the average of minus"

When we replace the fs with ps, we again notice that and so the average or expected value for is

Thus our completed formula for the variance of a discrete probability distribution is [in formula book]

We can now find the variance and standard deviation of the random variable in our example.

x	P(x)	xP(x)
0		0	0
1
2

Thus

Remember to square root this result to find the standard deviation, i.e.

 

Discrete Uniform Distributions

The uniform distribution occurs when all the outcomes in an experiment are equally likely. An example would be the outcome of the throw of a dice. The distribution would be:

The expectation and variance are simple to work out: