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Probability Mass Function (PMF)

Definition / Introduction

  • For a Discrete Random Variable (RV), we need a way to describe the probability associated with each specific value it can take.
  • The Probability Mass Function (PMF) provides this description. It gives the probability that a discrete random variable \(X\) is exactly equal to some specific value \(x\).
  • It essentially lists all possible numerical values the discrete RV can take and their corresponding probabilities.

Key Concepts

1. Notation and Definition

  • The PMF of a discrete random variable \(X\) is denoted by \(p(x)\), \(p_X(x)\), or \(P(X=x)\).
  • Definition: \(p(x) = P(X = x)\)
    • \(p(x)\) is the probability that the random variable \(X\) takes on the specific value \(x\).

2. Properties of a PMF

A function \(p(x)\) can be considered a valid PMF if and only if it satisfies these two conditions: * Non-negativity: The probability for every possible value \(x\) must be greater than or equal to zero. $$ p(x) \ge 0 \quad \text{for all possible values } x $$ * Summation to One: The sum of the probabilities for all possible values \(x\) that the random variable \(X\) can take must equal 1. This reflects that one of the possible outcomes must occur. $$ \sum_{\text{all } x} p(x) = 1 $$

3. Example: Fair Coin Flips

  • Experiment: Flip a fair coin twice.
  • Random Variable (X): Let X be the number of Heads. Possible values for X are \(\{0, 1, 2\}\).
  • Sample Space (S): \(\{TT, TH, HT, HH\}\) (Each outcome has probability \(1/4\)).
  • Calculating the PMF:
    • \(p(0) = P(X=0) = P(\{TT\}) = 1/4\)
    • \(p(1) = P(X=1) = P(\{TH, HT\}) = P(\{TH\}) + P(\{HT\}) = 1/4 + 1/4 = 2/4 = 1/2\)
    • \(p(2) = P(X=2) = P(\{HH\}) = 1/4\)

  • PMF Representation: We can represent this PMF as a table or a function:

    \(x\) (Value of X) \(p(x) = P(X=x)\)
    0 ¼
    1 ½
    2 ¼
    Total 1

  • Verification: Note that \(p(x) \ge 0\) for all \(x\), and \(\frac{1}{4} + \frac{1}{2} + \frac{1}{4} = 1\).

4. Visualization

  • A PMF is often visualized using a bar chart or a spike plot, where the height of the bar/spike at each value \(x\) represents the probability \(p(x)\).

5. Usage

  • The PMF allows us to calculate the probability of the RV falling within a certain range by summing the probabilities of the individual values within that range.
  • Example: Using the coin flip PMF, what is the probability of getting at least one Head?
    • \(P(X \ge 1) = P(X=1) + P(X=2) = p(1) + p(2) = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}\).

Connections to Other Topics

Summary

  • The PMF gives the probability \(P(X=x)\) for each possible value \(x\) of a discrete random variable \(X\).
  • Key properties: \(p(x) \ge 0\) and \(\sum p(x) = 1\).
  • It fully describes the probability distribution of a discrete RV.
  • Often visualized as a bar chart.

Sources