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CIE A-Level Maths Study Notes

4.5.2 Calculations Involving the Normal Distribution

In this section, the calculations involving a normally distributed variable, denoted as XN(μ,σ2)X \sim N(\mu, \sigma^2), where μ\mu is the mean and σ2\sigma^2 is the variance, will be explored in-depth. This exploration includes calculating probabilities, deriving relationships between variables, and the standardization process.

Basics of Normal Distribution

  • Normal distribution is a bell-shaped curve reflecting a continuous probability distribution.
  • Symmetry and Mean: The curve is symmetric around the mean, showing that values near the mean are more common.
  • Standard Deviation (σ)(\sigma): Determines the distribution's spread. A larger σ\sigma means a wider spread.
  • Real-World Examples: Common in heights, test scores, and measurement errors.

Probability Calculations in Normal Distribution

  • Calculating Probabilities: Focuses on the area under the curve.
  • Standard Normal Distribution Table: Shows the probability of a standard normal variable (Z)(Z) being within a range. It lists probabilities for values less than a given ZZ-score.

Example Problems

Example 1: Finding Probability (P(X > 12))

  • Standardize the Variable: Convert XX to ZZ using . Z=XμσZ = \frac{X - \mu}{\sigma} . For X=12,Z=1X = 12, Z = 1.
  • Calculate Probability: Find P(Z > 1) \approx 0.1587. This means P(X > 12) = 0.1587, or a 15.87% chance XX will be greater than 12.
Probability Graph

Example 2: Deriving Value of (x1)(x_1)

  • Find the ZZ-Score: For P(X < x_1) = 0.8413, ZZ-score is about 1.00.
  • Calculate x1x_1: Using x1=μ+Zσx_1 = \mu + Z\sigma, find x1=23x_1 = 23. There's an 84.13% chance XX will be less than 23.
Probability Graph

Example 3: Standardizing a Normal Variable

  • Calculate the ZZ-Score: For X=35X = 35 in N(30,16)N(30, 16), Z=1.25Z = 1.25.
  • Graphical Representation: Shows X=35X = 35 is 1.251.25 standard deviations above the mean of 30.
Probability Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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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