how do i calculate the mass function of a random variable XS

how do i calculate the mass function of a random variable X

Solution

The probability that a discrete random variable X takes on a particular value x, that is, P(X = x), is frequently denoted f(x). The function f(x) is typically called theprobability mass function, although some authors also refer to it as the probability function, the frequency function, or probability density function. We will use the common terminology — the probability mass function — and its common abbreviation —the p.m.f.

The probability mass function, P(X = x) = f(x), of a discrete random variable X is a function that satisfies the following properties:

(1) P(X = x) = f(x) > 0  if x the support S

(2)  xSf(x)=1xSf(x)=1

(3) P(XA)=xAf(x)

Item #1 basically says that, for every element x in the support S, all of the probabilities must be positive. Note that if x does not belong in the support S, then f(x) = 0. Item #2 basically says that if you add up the probabilities for all of the possible x values in the support S, then the sum must equal 1. And, item #3 says to determine the probability associated with the event A, you just sum up the probabilities of the x values in A.

Since f(x) is a function, it can be presented: