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CIE A-Level Maths Study Notes

5.3.2 Utilizing Probability Density Functions

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)0f(x) \geq 0 for all xx.
    • Normalization: Total area under f(x)f(x) over its range equals

Calculating Probabilities with PDFs

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

Examples

Example 1: Uniform Distribution

Question: For XX uniformly distributed between 0 and 1, find P(0.2X0.8)P(0.2 \leq X \leq 0.8).

Solution:

  • PDF: For XX uniform on [0, 1], f(x)=1f(x) = 1.
  • Probability: P(0.2X0.8)P(0.2 \leq X \leq 0.8) is the area under f(x)f(x) from 0.2 to 0.8.
  • Calculation: The area of a rectangle with height 1 over the interval 0.2,0.80.2, 0.8 is 0.80.2=0.60.8 - 0.2 = 0.6.

Result: P(0.2X0.8)=0.6.P(0.2 \leq X \leq 0.8) = 0.6.

Uniform Distribution Graph

Example 2: Normal Distribution

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

Solution:

  • Standard Normal PDF: f(x)=12πe12x2.f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}.
  • Find Probability: Integrate f(x)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%.

Normal Distribution Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

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.

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