1. Definition
- Probability mass refers to the probability assigned to a single outcome of a discrete random variable.
- Since discrete variables take only specific values (e.g., dice rolls, number of customers), each possible outcome gets a probability mass.
Formally:
If $X$ is a discrete random variable, then for any value $x$: $P(X = x) = p(x)$
Here $p(x)$ is the probability mass at point $x$.
2. Probability Mass Function (PMF)
- The PMF gives the probability mass for all possible values of the discrete variable.
- Must satisfy:
- $p(x) \geq 0$ for all $x$x.
- $\sum_x p(x) = 1$.
3. Examples
Example 1: Coin flip (Bernoulli)
- Outcomes: $X \in \{0,1\}$
- PMF:
- $p(1) = 0.5$ (heads)
- $p(0) = 0.5$ (tails)
Each outcome has probability mass = 0.5.
Example 2: Rolling a fair die
- Outcomes: $X \in \{1,2,3,4,5,6\}$
- PMF:
- $p(x) = 1/6$ for each $x$.
- Each outcome (e.g., rolling a 4) has probability mass = 1/6.
Example 3: Poisson Distribution (λ=2\lambda=2λ=2)
- Outcomes: $X \in \{0,1,2,\dots\}$
- PMF: $p(x) = \frac{e^{-\lambda}\lambda^x}{x!}$
- Probability mass decreases as $x$ gets large.
4. Contrast: Probability Mass vs. Probability Density
- Probability Mass = for discrete variables, exact value has probability > 0.
- Probability Density = for continuous variables, probability of an exact value = 0; probabilities come from integrating over intervals.
Summary:
- Probability mass = the probability assigned to a specific discrete outcome.
- Described by a PMF (Probability Mass Function).
- Used for discrete distributions like Bernoulli, Binomial, Poisson.