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Probability Distribution

Formula

\[ \text{Distribution of }X \text{ is the rule that gives } P(X \in A) \]

Parameters

  • \(X\): random variable
  • \(A\): event/set of values

What it means

A distribution specifies how probability mass or density is assigned across the possible values of a random variable.

What it's used for

  • Computing probabilities and expectations.
  • Choosing statistical models (e.g., Bernoulli, Normal, Poisson).

Key properties

  • Discrete distributions use a PMF.
  • Continuous distributions use a PDF (plus integration).
  • The CDF works for both discrete and continuous cases.

Common gotchas

  • A PDF value is not itself a probability.
  • The same family (e.g., Normal) can represent many distributions with different parameters.

Example

For a fair die roll \(X\in\{1,\dots,6\}\), the distribution is \(P(X=k)=1/6\) for each \(k\).

How to Compute (Pseudocode)

Input: random variable X or data/model assumptions
Output: probability distribution representation

choose a representation (PMF, PDF, CDF, parametric family, or empirical distribution)
estimate or specify the required parameters/rules
return the distribution object/representation

Complexity

  • Time: Depends on whether the distribution is specified analytically or estimated from data (empirical estimation is often linear in sample size)
  • Space: Depends on representation (parametric parameters vs histogram/empirical tables)
  • Assumptions: This is a modeling/representation workflow; downstream PMF/PDF/CDF computations determine concrete costs

See also