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 percentile, find the value where the area under the PDF to the left equals .
![Median and Percentile](https://tutorchase-production.s3.eu-west-2.amazonaws.com/999d682f-71f3-4635-98fe-102a01789342-file.png)
- 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: for .
- Median: For a uniform distribution, the median is the midpoint of the range.
- Computation: Median .
Conclusion: The median of the distribution is 5.
![Uniform Distribution Graph](https://tutorchase-production.s3.eu-west-2.amazonaws.com/4479bfe7-606b-4a04-a0ab-396a36603eea-file.jpg)
Example 2: Calculating the 25th Percentile
Determine the 25th percentile for a continuous random variable with PDF for .
Solution:
- 25th Percentile: The point where 25% of the values lie, or the CDF is 0.25.
- Integral Setup: Integrate from 0 to to find the CDF.
- Calculation: Solve .
- Solve for : Find when , which gives .
Conclusion: The 25th percentile of is 0.5.
![Distribution Graph](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F94ddd9cc-1848-4178-bda7-a30d7f6b2dd3-file.jpg&w=1080&q=75)