Normal Distribution (Further)

This lesson extends your understanding of the normal distribution to cover linear combinations of normal random variables, the distribution of sample means, and the foundations for inference. These results are essential for hypothesis testing and confidence intervals.

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Recap: The Normal Distribution

If $X \sim N(\mu, \sigma^2)$X∼N(μ,σ2):

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$f(x)=σ2π​1​e−2σ2(x−μ)2​

Standardisation: $Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$Z=σX−μ​∼N(0,1).

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Linear Combinations of Normal Variables

If $X \sim N(\mu_X, \sigma_X^2)$X∼N(μX​,σX2​) and $Y \sim N(\mu_Y, \sigma_Y^2)$Y∼N(μY​,σY2​) are independent:

Combination	Distribution
$aX + b$aX+b	$N(a\mu_X + b, \; a^2\sigma_X^2)$N(aμX​+b,a2σX2​)
$X + Y$X+Y	$N(\mu_X + \mu_Y, \; \sigma_X^2 + \sigma_Y^2)$N(μX​+μY​,σX2​+σY2​)
$X - Y$X−Y	$N(\mu_X - \mu_Y, \; \sigma_X^2 + \sigma_Y^2)$N(μX​−μY​,σX2​+σY2​)
$aX + bY$aX+bY	$N(a\mu_X + b\mu_Y, \; a^2\sigma_X^2 + b^2\sigma_Y^2)$N(aμX​+bμY​,a2σX2​+b2σY2​)

Critical point: For $X - Y$X−Y, the variances are added, not subtracted. This is the most common error in this topic.

Worked Example 1

The weight of a large egg is $X \sim N(68, 16)$X∼N(68,16) grams and a small egg is $Y \sim N(50, 9)$Y∼N(50,9) grams, independently.