1. Definition

  • For a continuous random variable, the probability density function (PDF) describes how probability is distributed across values.
  • The probability density at a point $x$, $f(x)$, is not the probability of $X=x$ (which is 0 for continuous variables).
  • Instead, it tells us how “dense” the probability is around $x$.

Formally: $P(a \leq X \leq b) = \int_a^b f(x)\, dx$

2. Properties of a PDF

  1. $f(x) \geq 0$ for all $x$.
  2. Total probability = 1:$\int_{-\infty}^{+\infty} f(x)\, dx = 1$
  3. Probability for a specific value = 0 (continuous variables).

3. Example: Normal Distribution

For $X \sim N(\mu, \sigma^2)$, the PDF is:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\!\Bigg(-\frac{(x-\mu)^2}{2\sigma^2}\Bigg)$

  • $f(x)$ is highest near $\mu$ (the mean).
  • To find probability of a range:
    • $P(\mu – \sigma \leq X \leq \mu + \sigma) = \int_{\mu – \sigma}^{\mu + \sigma} f(x)\,dx \approx 0.68$

4. Contrast with Probability Mass

  • Discrete random variable: Probability Mass Function (PMF).
    • $P(X=x) > 0$.
  • Continuous random variable: Probability Density Function (PDF).
    • $P(X=x) = 0$, but $P(a \leq X \leq b)$ is obtained by integrating density.

5. Examples

  • Uniform distribution [0,1]:
    • $f(x) = 1 \quad \text{for } 0 \leq x \leq 1$
    • So
      • $P(0.2 \leq X \leq 0.5) = \int_{0.2}^{0.5} 1 , dx = 0.3$
  • Exponential distribution ($\lambda=2$):
    • $f(x) = 2e^{-2x}, \; x \geq 0$
    • Then
      • $P(X \leq 1) = \int_0^1 2e^{-2x} dx = 1 – e^{-2} \approx 0.865$

Summary:

  • Probability density $f(x)$ describes how probability is distributed for a continuous random variable.
  • Probability of an interval is the area under the PDF curve.
  • Contrast: PMF for discrete variables vs PDF for continuous ones.