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.
Understanding Statistics: Probability Mass Functions
A PMF begins with a careful definition of the random variable. The individual outcomes of an experiment may be more detailed than the values recorded by the variable. For example, three coin tosses have eight equally likely sequences.
If X means the number of heads, the sequences HHT, HTH, and THH all produce the same value, two. This grouping is the key step. The value zero occurs once, one occurs three times, two occurs three times, and three occurs once.
Dividing each count by eight gives the probabilities. A PMF therefore depends on both the experiment and the rule used to turn outcomes into numbers.
Once the probabilities are known, they can be combined to answer practical questions. A teacher might record the number of absent students each day. A factory might record the number of faulty parts in a box.
If the concern is having at most one fault, add the probabilities for zero faults and one fault. This is often easier than finding every possible detailed outcome again. Expected value gives another useful summary.
For three fair coin tosses, the expected number of heads is one point five. No single set of three tosses has one point five heads.
The value describes the average from many repeated sets of tosses. It is a long-run center, not a prediction of one trial.
The shape of a PMF tells a story that one average cannot tell. Two processes can have the same expected value while having very different risks. Imagine two games with an average payout of five pounds.
One could pay five pounds every time. Another could usually pay nothing but occasionally pay a large amount. Their averages match, yet the experiences are very different.
The spread of values measures this difference. In statistics, variance describes spread by looking at how far values tend to lie from the expected value.
Larger spread means results are less predictable. When reading a PMF graph, notice where the tallest bars are, how wide the possible range is, and whether rare extreme values have meaningful probabilities.
Real data rarely provides exact probabilities at first. Students may estimate a PMF by repeating an experiment or collecting observations. If a bus arrival count of two passengers occurs on 18 out of 100 mornings, its estimated probability is 0.18.
Such estimates improve when more observations are collected, though they can still change over time if conditions change. Check the possible values before calculating anything. Counts cannot be negative or fractional.
Do not assume outcomes are equally likely unless the situation justifies it. A fair die has equal probabilities because its design aims for symmetry, while the number of customers entering a shop does not. Keep outcomes that are impossible at probability zero, and make sure every allowed value has been considered.
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.