What are E(Y) and Var(Y) for Y\simPo(\lambda)?

E(Y)=\lambda and Var(Y)=\lambda

Is the variance of a binomial variable larger or smaller than its mean?

Smaller, because Var(X)=np(1-p).

When can B(n,p) be approximated by a Poisson distribution?

When n is large and p is small.

Why does the Poisson approximation work for small p?

np(1-p)\approx np, so the variance is close to the mean.

Why is Po(6) a poor approximation to B(20,0.3)?

p is not small, and the variances differ: 4.2 against 6.

Mean, variance and Poisson approximation to the binomialEdexcel A-Level Further Maths: Flashcards

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What are $E(X)$ and $\text{Var}(X)$ for $X\sim B(n,p)$?

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What are $E(X)$ and $\text{Var}(X)$ for $X\sim B(n,p)$ ?: $E(X)=np$ , $\text{Var}(X)=np(1-p)$
What are $E(Y)$ and $\text{Var}(Y)$ for $Y\sim\text{Po}(\lambda)$ ?: $E(Y)=\lambda$ and $\text{Var}(Y)=\lambda$
Is the variance of a binomial variable larger or smaller than its mean?: Smaller, because $\text{Var}(X)=np(1-p)$ .
What does a Poisson variable's variance equal?: Its mean, $\lambda$ .
If a binomial has mean $\mu$ and variance $\sigma^2$ , what is $\frac{\sigma^2}{\mu}$ ?: $1-p$
How do you find $n$ once you know $p$ and the mean?: $n=\frac{\text{mean}}{p}$
When can $B(n,p)$ be approximated by a Poisson distribution?: When $n$ is large and $p$ is small.
What is the approximating distribution for $B(n,p)$ ?: $\text{Po}(np)$
$X\sim B(200,0.01)$ . Which Poisson distribution approximates it?: $\text{Po}(2)$
Why does the Poisson approximation work for small $p$ ?: $np(1-p)\approx np$ , so the variance is close to the mean.
Does the Poisson approximation overestimate or underestimate the variance?: Overestimates: $np>np(1-p)$ .
Why is $\text{Po}(6)$ a poor approximation to $B(20,0.3)$ ?: $p$ is not small, and the variances differ: $4.2$ against $6$ .

Exam questions on Mean, variance and Poisson approximation to the binomial

A box contains 40 light bulbs, each independently faulty with probability $0.15$ . The number of faulty bulbs in the box is $X$ , where $X\sim B(40,0.15)$ .
The random variable $Y\sim\text{Po}(\lambda)$ has the same mean as $X$ . Write down $\text{Var}(Y)$ and compare it with $\text{Var}(X)$ .2 marks
The random variable $X\sim B(n,p)$ has mean $12$ and variance $9.6$ .
Use a calculator to find $P(X>15)$ .2 marks
A factory makes pen cartridges. Each cartridge is independently defective with probability $0.004$ . Cartridges are packed in boxes of 500 and $X$ is the number of defective cartridges in a box.
Explain why $X$ may be approximated by a Poisson distribution and state the parameter of that distribution.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).