Understanding Probability Density Functions (PDFs) is vital in studying continuous random variables. Unlike discrete variables, continuous ones offer a broader scope of application, as they can assume any value within a given range. This segment explores how to apply PDFs for calculating probabilities over intervals and includes examples.

Continuous Random Variables & PDFs

• Continuous Random Variables: Can take any value within a range.
• Probability Density Function (PDF): Describes the likelihood of a continuous variable assuming a certain value.
• Properties of PDFs:
  • Non-Negativity: $f(x) \geq 0$ for all $x$.
  • Normalization: Total area under $f(x)$ over its range equals

Calculating Probabilities with PDFs

• Integral for Probability: To find probability $P(a \leq X \leq b)$, integrate the PDF from $a$ to $b$: $P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$

Examples

Example 1: Uniform Distribution

Question: For $X$ uniformly distributed between 0 and 1, find $P(0.2 \leq X \leq 0.8)$.

Solution:

• PDF: For $X$ uniform on [0, 1], $f(x) = 1$.
• Probability: $P(0.2 \leq X \leq 0.8)$ is the area under $f(x)$ from 0.2 to 0.8.
• Calculation: The area of a rectangle with height 1 over the interval $0.2, 0.8$ is $0.8 - 0.2 = 0.6$.

Result: $P(0.2 \leq X \leq 0.8) = 0.6.$

Example 2: Normal Distribution

Question: Given $X$ follows a standard normal distribution, what is $P(X < 1)$?

Solution:

• Standard Normal PDF: $f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}.$
• Find Probability: Integrate $f(x)$ from $-\infty$ to 1 to find $P(X < 1)$.
• Numerical Solution: Use standard normal tables or statistical software.

Result: $P(X < 1) \approx 0.8413$, or 84.13%.