The CDF looks like a somewhat weird way of describing the distribution of a random variable. A more natural idea is just assigning a probability to each possible outcome in the support. Unfortunately, in the next chapter we will see that this idea cannot be applied to a continuous random variable. Nevertheless, in the case of a discrete random variable we may indeed associate a probability with the event {X = xi}.
DEFINITION 6.4 (Probability Mass Function) The probability mass function (PMF) of a discrete random variable is defined as the function
The function is zero for values not included in the support. For values xi in the support, the shorthand notation pi ≡ P(X = xi) is often used.
Example 6.4 Consider the PMF in Table 6.1, and note that probabilities add up to 1. The PMF pX(x) is graphically illustrated in Fig. 6.2(a). The upward arrows are a common way to depict probability masses concentrated at discrete points within the distribution support. The height of each arrow corresponds to the probability of that value and provides us with a visual representation of likelihood. On the contrary, with continuous random variables the mass is distributed on continuous intervals. Figure 6.2(b) shows how we may get the CDF by summing up probabilities from the PMF:
Fig. 6.2 From PMF to CDF and vice versa.
Each probability we add over points in the support contributes a jump to the CDF. We can also go the other way around, i.e., we may obtain probabilities by taking differences in the CDF:
The example suggests general rules for moving from PMF and CDF and vice versa. We find the PMF by taking differences of consecutive values of the CDF over points xi in the support:
It is important to notice that if X is a discrete random variable and xi is in its support, then
Given the PMF, we just add up its values to find the CDF:
Indeed, the CDF and PMF provide us with the same information, which fully characterizes a discrete random variable.
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