Describe the shape of the Normal distribution.

Symmetrical and bell-shaped, centred on the mean.

How do you standardise X?

Z=\frac{X-\mu}{\sigma}\simN(0,1)

How do you find x from a probability?

Find z from the table, then x=\mu+z\sigma

How do you find \mu and \sigma from two probabilities?

Standardise each to give two equations in \mu and \sigma, then solve simultaneously

The Normal distributionEdexcel International A Level Maths: Flashcards

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Notation for a Normal distribution?

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Notation for a Normal distribution?: $X\sim\mathrm{N}(\mu,\sigma^2)$ , with mean $\mu$ and variance $\sigma^2$
Describe the shape of the Normal distribution.: Symmetrical and bell-shaped, centred on the mean.
$\mathrm{P}(X<\mu)$ for a Normal variable?: $0.5$
How do you standardise $X$ ?: $Z=\frac{X-\mu}{\sigma}\sim\mathrm{N}(0,1)$
What does $\Phi(z)$ mean?: $\mathrm{P}(Z<z)$
$\mathrm{P}(Z>z)$ in terms of $\Phi$ ?: $1-\Phi(z)$
$\mathrm{P}(Z<-z)$ in terms of $\Phi$ ?: $1-\Phi(z)$
$\mathrm{P}(-z<Z<z)$ in terms of $\Phi$ ?: $2\Phi(z)-1$
$z$ value for $\Phi(z)=0.95$ ?: $1.6449$
$z$ value for $\Phi(z)=0.90$ ?: $1.2816$
For $T\sim\mathrm{N}(42,9)$ , what is the standard deviation?: $3$ , since the variance is $9$
How do you find $x$ from a probability?: Find $z$ from the table, then $x=\mu+z\sigma$
How do you find $\mu$ and $\sigma$ from two probabilities?: Standardise each to give two equations in $\mu$ and $\sigma$ , then solve simultaneously

Exam questions on The Normal distribution

The mass of an egg, $X$ grams, is Normally distributed with mean 60 and standard deviation 4.
Find the probability that an egg has a mass between 54 g and 66 g.2 marks
The time, $T$ minutes, taken by a bus to complete its route is Normally distributed with $T\sim\mathrm{N}(42,\,9)$ .
Find the value of $t$ such that $\mathrm{P}(T<t)=0.9772$ .2 marks
The heights of adult women in a population are Normally distributed with mean 164 cm and standard deviation 6 cm.
Find the probability that a randomly chosen woman has a height between 158 cm and 173 cm.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).