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Probability Density Function (PDF)

Definition / Introduction

  • For a Continuous Random Variable, the probability of it taking on any single specific value is essentially zero (because there are infinitely many possible values in any interval).
  • Instead of assigning probabilities to specific points like a PMF does for discrete variables, we use a Probability Density Function (PDF) to describe the relative likelihood of a continuous random variable \(X\) falling within a given range or interval.
  • The PDF \(f(x)\) itself does not give probability directly; probability is represented by the area under the PDF curve over an interval.

Key Concepts

1. Notation and Definition

  • The PDF of a continuous random variable \(X\) is denoted by \(f(x)\) or \(f_X(x)\).
  • It's a function such that the probability of \(X\) falling within an interval \([a, b]\) is given by the integral (area under the curve) of \(f(x)\) from \(a\) to \(b\): $$ P(a \le X \le b) = \int_a^b f(x) \, dx $$

2. Properties of a PDF

A function \(f(x)\) can be considered a valid PDF if and only if it satisfies these two conditions: * Non-negativity: The density value must be greater than or equal to zero for all possible values \(x\). (Note: \(f(x)\) can be greater than 1, unlike a probability). $$ f(x) \ge 0 \quad \text{for all } x $$ * Total Area Equals One: The total area under the entire curve of the PDF must equal 1. This represents the certainty that the random variable will take some value within its possible range. $$ \int_{-\infty}^{\infty} f(x) \, dx = 1 $$

3. Key Difference from PMF

  • PMF (Discrete): \(p(x) = P(X=x)\). Gives direct probability at a point. Values are probabilities (\(0 \le p(x) \le 1\)). Summation \(\sum p(x) = 1\).
  • PDF (Continuous): \(f(x)\) is not \(P(X=x)\). \(P(X=x) = 0\). Gives probability density. Values \(f(x) \ge 0\) (can exceed 1). Integration \(\int f(x) dx = 1\). Probability is found by integrating \(f(x)\) over an interval \([a, b]\).

4. Example: Uniform Distribution

  • Consider a continuous random variable \(X\) uniformly distributed between 0 and 2, \(X \sim \text{Uniform}(0, 2)\). This means \(X\) is equally likely to fall anywhere in this interval.
  • PDF: $$ f(x) = \begin{cases} \frac{1}{2} & \text{if } 0 \le x \le 2 \ 0 & \text{otherwise} \end{cases} $$
  • Verification:
    • \(f(x) = 1/2 \ge 0\) within the range \([0, 2]\).
    • Total Area: \(\int_{-\infty}^{\infty} f(x) dx = \int_0^2 \frac{1}{2} \, dx = \left[ \frac{x}{2} \right]_0^2 = \frac{2}{2} - \frac{0}{2} = 1\).
  • Calculating Probability: What is the probability that X falls between 0.5 and 1.5?
    • \(P(0.5 \le X \le 1.5) = \int_{0.5}^{1.5} \frac{1}{2} \, dx = \left[ \frac{x}{2} \right]_{0.5}^{1.5} = \frac{1.5}{2} - \frac{0.5}{2} = \frac{1}{2} = 0.5\).
    • Visually, this is the area of a rectangle with width \((1.5 - 0.5) = 1\) and height \(1/2\).

5. Visualization

  • A PDF is visualized as a curve on a graph where the x-axis represents the values of the random variable and the y-axis represents the probability density \(f(x)\).
  • The area under the curve between two points represents the probability of the variable falling within that range.

Connections to Other Topics

Summary

  • The PDF describes the relative likelihood of values for a continuous random variable \(X\).
  • \(f(x)\) itself is not probability; area under the PDF curve represents probability. \(P(a \le X \le b) = \int_a^b f(x) dx\).
  • \(P(X=x) = 0\) for any specific \(x\) in a continuous distribution.
  • Key properties: \(f(x) \ge 0\) and \(\int_{-\infty}^{\infty} f(x) dx = 1\).
  • It fully describes the probability distribution of a continuous RV.

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