Properties of the probability mass functionΒΆ

There are some standard properties of the probability mass function that are worth memorizing.

First, he probability mass function is nonnegative:

$p(x) \ge 0,$

for the possible values $$x$$ that the random variable $$X$$ can take.

Second, the probability mass function is normalized:

$\sum_x p(x) = 1.$

The sum is over all the possible values of $$X$$. This is a direct consequence of the fact that $$X$$ must take some value of all that are possible.

Finally, you can use the probability mass function to find the probability that $$X$$ takes values in any set of possibilities. For example:

$p(X=x_1\;\text{or}\;X=x_2) = p(x_1) + p(x_2),$

if $$x_1\not= x_2$$.

Similarly, if $$x_1, x_2,$$ and $$x_3$$ are all different, then:

$p(X=x_1\;\text{or}\;X=x_2\;\text{or}\;X=x_3) = p(x_1) + p(x_2) + p(x_3).$

An, in general for any set of possible values of $$X$$, say $$A$$, the probability of $$X$$ taking values in $$A$$ is:

$p(X\in A) = \sum_{x\in A} p(x).$