The concept of determining medians and percentiles from Probability Density Functions (PDFs) is crucial for a comprehensive understanding of continuous random variables. This aspect of statistics enables us to quantify and interpret the behavior of a dataset or distribution, providing a deeper insight into its characteristics.

## Continuous Random Variables and PDFs

**Continuous Random Variables:**Can take infinite values within a range.**Probability Density Function (PDF):**A curve where the area under it between two points represents the probability of the variable falling within that range. Characteristics include:**Non-Negativity:**Always ≥ 0.**Total Area = 1:**Integral over its range equals 1.

## Medians and Percentiles

**Median:**The point dividing the distribution so 50% is to its left.**Calculation:**Find the value where the area under the PDF to the left is 0.5.

**Percentiles:**Points below which a certain percentage of data falls.**Calculation:**For the $p^{th}$ percentile, find the value where the area under the PDF to the left equals $\frac{p}{100}$.

- Image courtesy of Online Stat

## Example Problems

### Example 1: Finding the Median

Calculate the median of a uniform distribution where ( X ) is distributed evenly between 0 and 10.

**Solution**:

**PDF**: $f(x) = 0.1$ for $0 \leq x \leq 10$.**Median**: For a uniform distribution, the median is the midpoint of the range.**Computation**: Median $= \frac{0 + 10}{2} = 5$.

**Conclusion**: The median of the distribution is 5.

### Example 2: Calculating the 25th Percentile

Determine the 25th percentile for a continuous random variable $Y$ with PDF $f(y) = 2y$ for $0 \leq y \leq 1$.

**Solution**:

**25th Percentile**: The point where 25% of the values lie, or the CDF is 0.25.**Integral Setup**: Integrate $f(y) = 2y$ from 0 to $y$ to find the CDF.**Calculation**: Solve $\int_{0}^{y} 2y \, dy = 0.25$.**Solve for**$y$: Find $y$ when $y^2 = 0.25$, which gives $y = \sqrt{0.25} = 0.5$.

**Conclusion**: The 25th percentile of $Y$ is 0.5.

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.