FS1-2 Poisson and binomial distributions — 9FM0 spec coverage

FS1-2.1

The Poisson distribution. The additive property of Poisson distributions.

A Poisson model counts events occurring independently at a constant mean rate, with $P(X=x)=e^{-\lambda}\dfrac{\lambda^x}{x!}$ for $x=0,1,2,\ldots$ .
Scale $\lambda$ with the length of the interval, and add parameters for independent Poisson counts: $X+Y\sim\operatorname{Po}(\lambda+\mu)$ .
For example, a rate of $1.6$ events per hour gives a mean of $4$ events in $2.5$ hours.
A common error is to add Poisson parameters for overlapping intervals or dependent counts; the additive result requires independence.

Tier 1 · Easy

2 marks

ORIGINAL

A sensor records a Poisson-distributed number of amber flashes with mean $2.7$ per minute. Find the probability of no amber flashes in one minute.

Tier 2 · Standard

3 marks

ORIGINAL

Independent counts $A$ and $B$ have distributions $\operatorname{Po}(3.2)$ and $\operatorname{Po}(1.7)$ . Find $P(A+B\geq7)$ .

Tier 3 · Hard

8 marks

ORIGINAL

Inspection pings are modelled by a Poisson process at a mean rate of $0.8$ per $10$ minutes. Find the probability of exactly $2$ pings in the first $15$ minutes and at most $2$ pings in the next $30$ minutes. State two assumptions needed for the model and one feature of the real process that would make it unsuitable.

FS1-2.2

The mean and variance of the binomial distribution and the Poisson distribution.

If $X\sim\operatorname{Bin}(n,p)$ , then $E(X)=np$ and $\operatorname{Var}(X)=np(1-p)$ .
If $Y\sim\operatorname{Po}(\lambda)$ , then both $E(Y)$ and $\operatorname{Var}(Y)$ equal $\lambda$ .
For a linear transformation, $E(aX+b)=aE(X)+b$ and $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$ .
A common error is to use $np$ as the binomial variance or to add the constant $b$ when transforming a variance.

Tier 1 · Easy

2 marks

ORIGINAL

Given $X\sim\operatorname{Bin}(80,0.15)$ , find $E(X)$ and $\operatorname{Var}(X)$ .

Tier 2 · Standard

4 marks

ORIGINAL

The random variable $Y$ is Poisson and $\operatorname{Var}(2Y+3)=28$ . Find the parameter of $Y$ and $E(2Y+3)$ .

Tier 3 · Hard

5 marks

ORIGINAL

A binomial random variable $X$ has mean $18$ and variance $13.5$ . Determine $n$ and $p$ .

FS1-2.3

The use of the Poisson distribution as an approximation to the binomial distribution.

When $n$ is large and $p$ is small, $\operatorname{Bin}(n,p)$ can be approximated by $\operatorname{Po}(np)$ .
Keep the probability event unchanged and use $\lambda=np$ ; a continuity correction is not used for a binomial-to-Poisson approximation.
For example, $\operatorname{Bin}(900,0.004)$ is approximated by $\operatorname{Po}(3.6)$ .
A common error is to quote only $np$ without checking that $n$ is large and $p$ is small.

Tier 1 · Easy

3 marks

ORIGINAL

Let $X\sim\operatorname{Bin}(900,0.004)$ . State a suitable Poisson approximation and use it to estimate $P(X=0)$ .

Tier 2 · Standard

4 marks

ORIGINAL

The number of flawed seals in a batch has distribution $X\sim\operatorname{Bin}(500,0.008)$ . Use a Poisson approximation to estimate $P(X\geq6)$ .

Tier 3 · Hard

6 marks

ORIGINAL

For $X\sim\operatorname{Bin}(200,0.015)$ , calculate $P(X\leq2)$ exactly and by a Poisson approximation. Find the percentage error of the approximation relative to the exact value.