Probability Mass

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 $x$.

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2. Probability Mass Function (PMF)

• The PMF gives the probability mass for all possible values of the discrete variable.
• Must satisfy:
  1. $p(x) \geq 0$ for all $x$x.
  2. $\sum_x p(x) = 1$.

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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.

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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.

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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.

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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.

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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.