Probability Density Function (PDF)

Definition

The Probability Density Function (PDF) describes how likely a continuous random variable $ X $ takes on a specific value.

For a random variable $ X $, the PDF is denoted as:

\[f(x) = \frac{d}{dx} F(x)\]

where $ F(x) $ is the Cumulative Distribution Function (CDF).

Key Properties of PDF

  1. Non-negativity: $ f(x) \geq 0 $ for all $ x $.

  2. Total area under the curve is 1:

    \[\int_{-\infty}^{\infty} f(x) dx = 1\]
  3. It does not directly give probability of a single value, but rather how densely probability is distributed.

Example: Normal Distribution PDF

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

\[f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)\]
  • $ \mu $ = mean (center of the distribution).
  • $ \sigma $ = standard deviation (spread of data).

Visual Interpretation

  • The PDF graph shows where values are most likely to occur.
  • In a normal distribution, the peak is at $ x = \mu $ (mean), and probability decreases as you move away.

2. Cumulative Distribution Function (CDF)

Definition

The Cumulative Distribution Function (CDF) gives the probability that a random variable $ X $ takes on a value less than or equal to $ x $:

\[F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt\]

Key Properties of CDF

  1. Monotonic increasing: $ 0 \leq F(x) \leq 1 $.
  2. Limits:
    • $ F(-\infty) = 0 $ (probability is 0 before any values occur).
    • $ F(\infty) = 1 $ (eventually, all probability accumulates to 1).
  3. Deriving PDF from CDF:
    • The PDF is the derivative of the CDF:
      \(f(x) = \frac{d}{dx} F(x)\)

Example: Normal Distribution CDF

For a normal distribution $ X \sim N(\mu, \sigma^2) $, the CDF is:

\[F(x) = \frac{1}{2} \left[1 + \text{erf} \left(\frac{x - \mu}{\sqrt{2\sigma^2}} \right) \right]\]

where erf() is the error function.

Relationship Between PDF and CDF

| Concept | Probability Density Function (PDF) | Cumulative Distribution Function (CDF) | |———|——————————–|——————————–| | Definition | Describes probability density at a point $ x $. | Gives probability that $ X $ is less than or equal to $ x $. | | Mathematical Relation | $ f(x) = \frac{d}{dx} F(x) $ | $ F(x) = \int_{-\infty}^{x} f(t) dt $ | | Graph Shape | Bell-shaped for normal distributions, can vary for others. | Always monotonically increasing from 0 to 1. | | Use Case | Helps understand the most likely values. | Helps compute probabilities of intervals. |

5. Why Are PDF and CDF Important?

  • PDF is useful for modeling distributions and determining the likelihood of specific values.
  • CDF is essential for probability calculations, such as:
    • Finding the probability of intervals: $ P(a \leq X \leq b) = F(b) - F(a) $.
    • Setting percentiles: The median is where $ F(x) = 0.5 $.

Understanding PDF and CDF is fundamental in Extreme Value Theory (EVT) since:

  • PDF helps model how extreme values are distributed (e.g., tail behavior).
  • CDF helps compute exceedance probabilities, which are critical in risk assessment (e.g., probability of a flood exceeding a certain height).

By mastering these concepts, you build a strong foundation for working with probability distributions, EVT, and GPD modeling!