PDF

PDF (Probability Density Function) is a function that describes the relative likelihood of a continuous random variable taking on a specific value. Here are the key points about PDFs:

1. Mathematical notation: For a random variable $X$, the PDF is often denoted as $f_X(x)$ or just $f(x)$
2. Properties:
   • Non-negative: $f(x)\ge 0$ for all $x$
     • Probability is always non-negative in real world
     • Negative probability doesn’t make physical sense
   • Total area equals 1: ${\int }_{-\infty }^{\infty }f(x)dx=1$
     • The integral represents total probability across all possible values
     • By axioms of probability, total probability must equal 1
     • If area was > 1, we’d have >100% probability
     • If area was < 1, we’d have missing probability
     • This property ensures probabilities are properly normalized
3. Probability calculation:
   • For an interval $[a,b]$, probability is: $P(a\le X\le b)={\int }_a^{b}f(x)dx$
   • For a single point: $P(X=x)=0$ for continuous distributions
4. Common examples:
   • Normal distribution: $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$
   • Uniform distribution: $f(x)=\frac{1}{b-a}$ for $x\in [a,b]$

The PDF is different from PMF (Probability Mass Function), which is used for discrete random variables.