Probability Distributions

In the following P(x) represents the probability of x.

Assume that x is a random variable that assumes the values of 0, 1, 2, and 3.

Note that for each x,

Further note that .3 + .25 + .3 + .15 = 1

Using sigma notation, the short hand for this last sum is

A probability distribution must satisfy each of the following:

(where x assumes all possible values)

and

(for each value of x)

Example 1

Verify that the following is a probability distribution. Assume that x only takes on the values 0, 1, and 2.

Solution .

Since each P(x) is nonnegative and since .25 + .5 + .25 = 1, we do have a probability distribution.

In fact this probability distribution arises when you consider the experiment of tossing a fair coin twice and recording the number of heads. You will either get 0, 1, or 2 heads with the probabilities indicated in the table in example one.

Example 2

Does the following give a probability distribution?

Solution .

Since .3 + .2 + .1 + .2 = .8, this is not a probability distribution.

Example 3

Assume that x can take on only the values of 0 and 1. Does P(x) = (x + 1)/3 give a probability distribution?

Solution .

Note that P(0) = (0+1)/3 = 1/3 and P(1) = (1+1)/3 = 2/3. Since 1/3 + 2/3 = 3/3 = 1, this is a probability distribution.