Discrete random variables and probability distributions

What the exam asks

• Identify or classify a variable as a discrete random variable and verify whether a given table, graph, or function is a valid probability distribution.
• Calculate missing probabilities using the rule that all probabilities in a distribution must sum to $1$.
• Construct or complete a probability distribution from probability rules (such as independence or multiplication rules) or estimate one from simulation results.
• Calculate and interpret probabilities like $P(X > x)$, $P(X \ge x)$, or $P(a \le X \le b)$ from a standard or cumulative probability distribution.
• Describe the shape, center, variability, and unusual features of a discrete probability distribution shown as a histogram.

Core ideas

Random variables and discrete distributions

A random variable assigns a unique numerical value to each outcome of a random phenomenon. We use a capital letter (like $X$) for the random variable and a lowercase letter (like $x_i$) for a particular value it can take.

A discrete random variable takes a countable number of possible values (often whole numbers like $0, 1, 2, \dots$, though decimals like shoe sizes $7.5, 8.0, 8.5$ are also countable).

A probability distribution of a discrete random variable $X$ gives every possible value $x_i$ and its probability $P(X = x_i)$ (often written ). It can be displayed as a table, a histogram, or a formula/function. Every valid discrete probability distribution must satisfy two conditions:

1. Every probability is between $0$ and $1$ inclusive: $0 \le P(X = x_i) \le 1$.

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