NSW Y12 Maths - Advanced Random Variables The Normal Distribution

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The Normal Distribution Theory

Approximately \(68 \%\) of the distributions will lie within one standard deviation \((\sigma)\) of the mean \((\mu)\).\\  Approximately 95\% of the distributions will lie within two standard deviations of the mean.\\  Approximately \(99.7 \%\) of the distributions mill line within three standard deviations of the mean.\\  A continuous random variable that has a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is written as \(X-N\left(\mu, \sigma^2\right)\)\\  \textbf{Example 1}\\ \(X-N(12,4)\) what is the range of values you would expect to find in: \\ (i) the middle \(68 \%\)\\ (ii) the middle \(95 \%\)\\ (iv) the middle \(99.7 \%\)\\  \textbf{Example 1 solution}\\ (i) \(\sigma^2=4 \rightarrow \sigma=2\). \[ \begin{aligned} \therefore 12-2<x & <12+2 \\ 10 & <x<14 \end{aligned} \] (ii) \(12-2 \times 2<x<12+2 \times 2\) \(8<x<16\)\\  (iii) \(12-3 \times 2<x<12+3 \times 2\) \[ 6<x<18 \]

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  • The Normal Distribution - Video - The normal distribution Introduction

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