When is a normal approximation to B(n,p) suitable?

When n is large and p is close to 0.5 (a check is np5 and nq5).

When is a normal approximation to Po(\lambda) suitable?

When \lambda is large, usually \lambda10.

Normal approximation to X\simB(n,p)?

X\approxN(np,\ npq) where q=1-p.

What is a continuity correction?

An adjustment of \pm0.5 that allows for a discrete variable being approximated by a continuous one.

P(X<k) with a continuity correction?

P(Y<k-0.5), because X<k means X\le k-1.

P(Xk) with a continuity correction?

P(Yk+0.5), because Xk means X\ge k+1.

What is the second parameter in N(\mu,\sigma^2)?

The variance, not the standard deviation.

Why is N(\lambda,\lambda) a natural approximation for a Poisson distribution?

A Poisson distribution has mean \lambda and variance \lambda, and the normal distribution is matched to both.

How do you find the smallest integer k with P(X\le k)\ge p?

Solve \frac{k+0.5-\mu}{\sigma}\ge z using the percentage point, then round k up to an integer.

Normal approximation to the binomial and PoissonEdexcel International A Level Further Maths: Flashcards

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When is a normal approximation to $\mathrm{B}(n,p)$ suitable?

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When is a normal approximation to $\mathrm{B}(n,p)$ suitable?: When $n$ is large and $p$ is close to $0.5$ (a check is $np>5$ and $nq>5$ ).
When is a normal approximation to $\mathrm{Po}(\lambda)$ suitable?: When $\lambda$ is large, usually $\lambda>10$ .
Normal approximation to $X\sim\mathrm{B}(n,p)$ ?: $X\approx\mathrm{N}(np,\ npq)$ where $q=1-p$ .
Normal approximation to $X\sim\mathrm{Po}(\lambda)$ ?: $X\approx\mathrm{N}(\lambda,\ \lambda)$ .
What is a continuity correction?: An adjustment of $\pm0.5$ that allows for a discrete variable being approximated by a continuous one.
$\mathrm{P}(X=k)$ with a continuity correction?: $\mathrm{P}(k-0.5<Y<k+0.5)$ .
$\mathrm{P}(X\le k)$ with a continuity correction?: $\mathrm{P}(Y<k+0.5)$ .
$\mathrm{P}(X\ge k)$ with a continuity correction?: $\mathrm{P}(Y>k-0.5)$ .
$\mathrm{P}(X<k)$ with a continuity correction?: $\mathrm{P}(Y<k-0.5)$ , because $X<k$ means $X\le k-1$ .
$\mathrm{P}(X>k)$ with a continuity correction?: $\mathrm{P}(Y>k+0.5)$ , because $X>k$ means $X\ge k+1$ .
What is the second parameter in $\mathrm{N}(\mu,\sigma^2)$ ?: The variance, not the standard deviation.
Why is $\mathrm{N}(\lambda,\lambda)$ a natural approximation for a Poisson distribution?: A Poisson distribution has mean $\lambda$ and variance $\lambda$ , and the normal distribution is matched to both.
How do you find the smallest integer $k$ with $\mathrm{P}(X\le k)\ge p$ ?: Solve $\frac{k+0.5-\mu}{\sigma}\ge z$ using the percentage point, then round $k$ up to an integer.

Exam questions on Normal approximation to the binomial and Poisson

The random variable $X$ has distribution $\mathrm{B}(80,0.4)$ , and a normal approximation is to be used.
Hence calculate an approximation to $\mathrm{P}(X\le30)$ .2 marks
The number of emails $X$ received by a help desk in one hour has distribution $\mathrm{Po}(25)$ , and a normal approximation is to be used.
Hence calculate an approximation to $\mathrm{P}(X\ge30)$ .2 marks
A component is defective with probability $0.45$ , independently of other components. In a random sample of $150$ components, $X$ is the number that are defective.
Using a suitable normal approximation, find $\mathrm{P}(X=70)$ .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).