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CIE A-Level Maths Study Notes

5.2.3 Normal Distribution of Linear Combinations

The concept of linear combinations of normally distributed variables is a significant topic in probability and statistics. This section delves into how these combinations maintain their normal distribution, an essential aspect of understanding real-world data distributions.

Understanding Normal Distribution

  • Normal Distribution: A bell-shaped curve described by its mean μμ and variance σ2σ², used to model various real-world phenomena.

Linear Combinations in Normal Distribution

  • Linear Combination: The process of adding variables together, each multiplied by a constant. It's a foundation in statistics for modeling and analysis.

Key Principle

  • Linear Combinations of Normally Distributed Variables: If the variables are normally distributed, their linear combinations also follow a normal distribution.
  • Single Variable: If XN(μ,σ2)X \sim N(\mu, \sigma^2), then aX+baX + b is also normally distributed as N(aμ+b,a2σ2N(a\mu + b, a^2\sigma^2.
  • Two Independent Variables: For XN(μ1,σ12)X \sim N(\mu_1, \sigma_1^2) and YN(μ2,σ22)Y \sim N(\mu_2, \sigma_2^2), aX+bYaX + bY follows N(aμ1+bμ2,a2σ12+b2σ22)N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2).

Examples

Example 1: Single Variable Transformation

Question: Given a normal distribution N(10,4)N(10, 4) for a random variable XX, determine the distribution of the transformed variable 3X+23X + 2.

Solution:

1. Mean of (3X + 2):

Calculated using E(3X+2)=3E(X)+2E(3X + 2) = 3E(X) + 2.

With E(X)=10E(X) = 10, it results in E(3X+2)=3×10+2=32E(3X + 2) = 3 \times 10 + 2 = 32.

2. Variance of (3X + 2):

Determined by Var(3X+2)=32Var(X)\text{Var}(3X + 2) = 3^2 \text{Var}(X).

Given Var(X)=4\text{Var}(X) = 4, this leads to Var(3X+2)=9×4=36\text{Var}(3X + 2) = 9 \times 4 = 36.

Conclusion: The transformed variable 3X+23X + 2 follows a normal distribution N(32,36)N(32, 36).

Single Variable Transformation Graph

Example 2: Combination of Two Independent Variables

Question: What is the distribution of the variable 2XY2X - Y if XX and YY are independent normally distributed variables with XN(5,9)X \sim N(5, 9) and YN(3,16)Y \sim N(3, 16)?

Solution:

1. Mean: The mean of 2XY2X - Y is 2×53=72 \times 5 - 3 = 7.

2. Variance: The variance of 2XY2X - Y is 22×9+(1)2×16=52.2^2 \times 9 + (-1)^2 \times 16 = 52.

Conclusion: 2XY2X - Y is normally distributed with N(7,52)N(7, 52).

Combination of Two Independent Variables 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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