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%.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.