Probability mass function (PMF) :
PMF for a discrete random variable assigns probabilities to each possible outcome of an experiment. Specifically, it determines the probability that the random variable X will equal a particular value xi, denoted as P(X = xi). In simpler terms, the PMF is a mathematical function that connects the individual outcomes of a random event to their respective probabilities.
Let us take one simple example to understand it easily.
Consider the daily frequency of fraudulent transactions at a bank branch and the associated probabilities presented in the table below. The table shows various potential values of the random variable, representing the number of fraudulent transactions and their respective probabilities.
From the table, we have the following information.
The probability that there will be no fraudulent transaction on any given day,
P(X = 0) = 0.20.
Similarly, P(X = 1) = 0.15, P(X = 2) = 0.25, P(X = 3) = 0.25 etc.
The probability mass function, P(xi), satisfies the following conditions:
P(xi) >= 0
Sum of all P(xi) = 1
Cumulative distribution function (CDF):
F(xi) is the probability that the random variable X takes values less than or equal to xi.
That is, F(xi) = P(X ≤ xi).
From the table above,
F(2) = P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.60
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