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PDF (Probability Density Function)

Formula

\[ P(a \le X \le b)=\int_a^b f_X(x)\,dx \]
\[ f_X(x)\ge 0,\quad \int_{-\infty}^{\infty} f_X(x)\,dx=1 \]

Parameters

  • \(X\): continuous random variable
  • \(f_X(x)\): density at value \(x\)

What it means

A PDF describes how probability is distributed over a continuous variable. Probabilities come from area under the curve, not point values.

What it's used for

  • Computing interval probabilities by integration.
  • Defining continuous models such as the Normal distribution.

Key properties

  • Probabilities are areas under \(f_X(x)\).
  • \(P(X=x)=0\) for continuous \(X\) (under standard continuous models).

Common gotchas

  • \(f_X(x)\) can be greater than 1; that is still valid if total area is 1.
  • The PDF itself is not a probability at a point.

Example

If \(X\sim \mathrm{Uniform}(0,1)\), then \(f_X(x)=1\) on \([0,1]\), so \(P(0.2\le X\le 0.5)=0.3\).

How to Compute (Pseudocode)

Input: distribution specification and query value(s)
Output: PDF values (or probabilities from it)

for each query value/interval:
  evaluate the distribution rule for the card's representation
  (PMF: point probability, PDF: density, CDF: cumulative probability)
return the computed value(s)

Complexity

  • Time: Depends on the distribution family and number of queries (often \(O(q)\) for \(q\) query points once parameters are known)
  • Space: \(O(q)\) for returned values (or \(O(1)\) for a single query)
  • Assumptions: Parameter-estimation cost is excluded; exact formulas and numerical methods vary by distribution family

See also