A probability mass function, or PMF, describes the probabilities of all possible values of a discrete random variable. It is used when outcomes can be counted, such as dice rolls, number of heads, or number of defective items. A PMF matters because it turns a random process into a clear table, formula, or bar chart that can be analyzed.
Each bar in the chart shows how likely one specific outcome is.
Key Facts
- A PMF assigns a probability to each possible value of a discrete random variable.
- For every outcome x, 0 ≤ P(X = x) ≤ 1.
- All probabilities in a PMF must add to 1: Σ P(X = x) = 1.
- For a fair six-sided die, P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6.
- Expected value is the long-run average: E(X) = Σ xP(X = x).
- For any event A, P(X is in A) = Σ P(X = x) over all x in A.
Vocabulary
- Probability Mass Function
- A probability mass function gives the probability that a discrete random variable equals each possible value.
- Discrete Random Variable
- A discrete random variable is a variable whose possible values are separate countable outcomes.
- Outcome
- An outcome is one possible result of a random process, such as rolling a 4 on a die.
- Expected Value
- Expected value is the weighted average of a random variable using probabilities as weights.
- Probability Distribution
- A probability distribution describes how probability is assigned across all possible outcomes of a random variable.
Common Mistakes to Avoid
- Forgetting that probabilities must sum to 1. If the total is less than or greater than 1, the list cannot be a valid PMF.
- Assigning a negative probability to an outcome. Probabilities cannot be negative because they represent portions of total likelihood.
- Confusing a PMF with a histogram of data counts. A PMF gives theoretical or modeled probabilities, while a data histogram shows observed frequencies from a sample.
- Adding x-values instead of probabilities when finding an event probability. To find P(X is in A), add the probabilities for the relevant outcomes, not the outcomes themselves.
Practice Questions
- 1 A random variable X has PMF P(X = 0) = 0.2, P(X = 1) = 0.5, and P(X = 2) = 0.3. Find P(X ≥ 1) and E(X).
- 2 A PMF is given by P(X = 1) = 0.15, P(X = 2) = 0.25, P(X = 3) = k, and P(X = 4) = 0.35. Find k, then find P(X is even).
- 3 A student draws a bar chart for a PMF and the bars have heights 0.1, 0.2, 0.4, and 0.5. Explain whether this can represent a valid PMF and justify your answer.