A probability mass function (PMF) is a function that describes the probability distribution of a discrete random variable. It assigns probabilities to each possible value that the random variable can take on.
For a given discrete random variable X, the probability mass function P(X = x) gives the probability of X taking on the value x. The PMF satisfies the following properties:
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Non-negativity: The probability assigned by the PMF is non-negative for all possible values of X.
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Summation: The sum of probabilities over all possible values of X is equal to 1. In mathematical notation, ∑ P(X = x) = 1, where the sum is taken over all possible values of x.
The probability mass function allows us to determine the likelihood of different outcomes of a discrete random variable. By evaluating the PMF at specific values, we can calculate the probability of observing those values.
For example, let's consider a fair six-sided die. The random variable X represents the outcome of rolling the die. The PMF for this random variable would be:
P(X = 1) = 1/6 P(X = 2) = 1/6 P(X = 3) = 1/6 P(X = 4) = 1/6 P(X = 5) = 1/6 P(X = 6) = 1/6
These probabilities represent the equal chance of obtaining each outcome when rolling a fair six-sided die.
The PMF is a fundamental concept in probability theory and is widely used in various fields, including statistics, discrete mathematics, and computer science. It provides a concise way to represent the probability distribution of discrete random variables and is crucial for analyzing and understanding random phenomena with discrete outcomes.