3.5 Properties of pmfs and cdfs

Let’s define the support of a random variable $$X$$ — this is the set of all possible values that $$X$$ can possibly take. We’ll use the notation $$\mathcal{X}$$ to denote the support of $$X$$.

Example 3.5 Suppose $$X$$ is the outcome from a roll of a die. Then, the support of $$X$$ is given by $$\mathcal{X} = \{1,2,3,4,5,6\}$$. In other words, the only possible values for $$X$$ are from $$1,\ldots,6$$.

Example 3.6 Suppose $$X$$ is the number of years of education that a person has. The support of $$X$$ is given by $$\mathcal{X} = \{0, 1, 2, \ldots, 20\}$$. Perhaps I should have chosen a larger number than 20 to be the maximum possible value that $$X$$ could take, but you will get the idea — a person’s years of education can be 0 or 1 or 2 or up to some maximum value.

Properties of pmfs

1. For any $$x$$, $$0 \leq f_X(x) \leq 1$$

In words: the probability of $$X$$ taking some particular value can’t be less than 0 or greater than 1 (neither of those would make any sense)

2. $$\sum_{x \in \mathcal{X}} f_X(x) = 1$$

In words: if you add up $$\mathrm{P}(X=x)$$ across all possible values that $$X$$ could take, they sum to 1.

Properties of cdfs for discrete random variables

1. For any $$x$$, $$0 \leq F_X(x) \leq 1$$

In words: the probability that $$X$$ is less than or equal to some particular value $$x$$ has to be between 0 and 1.

2. If $$x_1 < x_2$$, then $$F_X(x_1) \leq F_X(x_2)$$

In words: the cdf is increasing in $$x$$ (e.g., it will always be the case that $$\mathrm{P}(X \leq 3) \leq \mathrm{P}(X \leq 4)$$).

3. $$F_X(-\infty)=0$$ and $$F_X(\infty)=1$$

In words: if you choose small enough values of $$x$$, the probability that $$X$$ will be less than that is 0; similar (but opposite) logic applies for big values of $$x$$.

Connection between pmfs and cdfs

1. $$F_X(x) = \displaystyle \sum_{z \in \mathcal{X} \\ z \leq x} f_X(z)$$

In words: you can “recover” the cdf from the pmf by adding up the pmf across all possible values that the random variable could take that are less than or equal to $$x$$. This will be clearer with an example:

Example 3.7 Suppose that $$X$$ is the outcome of a roll of a die. Earlier we showed that $$F_X(3) = 1/2$$. We can calculate this by

\begin{aligned} F_X(3) &= \sum_{\substack{z \in \mathcal{X} \\ z \leq 3}} f_X(z) \\ &= \sum_{z=1}^3 f_X(z) \\ &= f_X(1) + f_X(2) + f_X(3) \\ &= \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \\ &= \frac{1}{2} \end{aligned}