Poisson approximation to binomial | CAIE 9709 Probability & Statistics 2 Notes

Chapters

Probability & Statistics 2

Mathematics9709

Check the conditions and preserve the mean

Many trials + rare success

X ~ B(n, p)

n > 50 and np < 5

keep the mean: λ = np ↓

X ≈ Po(np)

state this distribution before calculating

No continuity correction: both models are discrete.

The Poisson approximation keeps the binomial mean: λ = np.

B(n,p)\approx\operatorname{Po}(np)

Symbols

$n$ = number of independent trials (none)
$p$ = small success probability (none)
$np$ = Poisson parameter (successes)

Cambridge's guide is $n>50$ and $np<5$ , with p small.
State the approximating distribution before calculating.
Do not use a continuity correction: both distributions are discrete.

Turn a rare-success problem into Poisson

Worked example

Current-paper wind-turbine sample

One percent of houses have a wind turbine. A random sample of 400 houses is taken. Approximate $P(X\le3)$ .

X\sim B(400,0.01)\approx\operatorname{Po}(4)

S=1+4+\frac{4^2}{2!}+\frac{4^3}{3!}

P(X\le3)=e^{-4}S

Answer

P(X\le3)=0.433

(9709/61/M/J/25 Q1)

Key idea

The approximation changes the model, not the event. An event such as

X<4

still means the integer counts 0, 1, 2 and 3.

Use the zero-count shortcut in reverse

For a Poisson variable, $P(X=0)=e^{-\lambda}$ . This often turns an unknown-rate question into one logarithm.

Worked example

Current-paper reduced error rate

A data set contains 14 500 characters and the error probability is initially 0.0001. Find the new error probability p if the chance of no errors is doubled.

\lambda_0=14500(0.0001)=1.45

e^{-14500p}=2e^{-1.45}

14500p=1.45-\ln2

Answer

p=0.0000522\ \text{(3 s.f.)}

(9709/61/O/N/25 Q3(b))

Your turnIndependent Poisson-approximation check [3 marks]

$X\sim B(1000,0.004)$ . Use a suitable approximation to find $P(X>5)$ .

Show worked answer

X\approx\operatorname{Po}(4)

P(X>5)=1-P(X\le5)

Answer

P(X>5)=0.215

Common mistake

Poisson has one parameter. Write

\operatorname{Po}(np)

, not

\operatorname{Po}(n,p)

Check the conditions and preserve the mean

Many trials + rare success

X ~ B(n, p)

n > 50 and np < 5

keep the mean: λ = np ↓

X ≈ Po(np)

state this distribution before calculating

No continuity correction: both models are discrete.

The Poisson approximation keeps the binomial mean: λ = np.

B(n,p)\approx\operatorname{Po}(np)

Symbols

$n$ = number of independent trials (none)
$p$ = small success probability (none)
$np$ = Poisson parameter (successes)

Cambridge's guide is $n>50$ and $np<5$ , with p small.
State the approximating distribution before calculating.
Do not use a continuity correction: both distributions are discrete.

Turn a rare-success problem into Poisson

Worked example

Current-paper wind-turbine sample

One percent of houses have a wind turbine. A random sample of 400 houses is taken. Approximate $P(X\le3)$ .

X\sim B(400,0.01)\approx\operatorname{Po}(4)

S=1+4+\frac{4^2}{2!}+\frac{4^3}{3!}

P(X\le3)=e^{-4}S

Answer

P(X\le3)=0.433

(9709/61/M/J/25 Q1)

Key idea

The approximation changes the model, not the event. An event such as

X<4

still means the integer counts 0, 1, 2 and 3.

Use the zero-count shortcut in reverse

For a Poisson variable, $P(X=0)=e^{-\lambda}$ . This often turns an unknown-rate question into one logarithm.

Worked example

Current-paper reduced error rate

A data set contains 14 500 characters and the error probability is initially 0.0001. Find the new error probability p if the chance of no errors is doubled.

\lambda_0=14500(0.0001)=1.45

e^{-14500p}=2e^{-1.45}

14500p=1.45-\ln2

Answer

p=0.0000522\ \text{(3 s.f.)}

(9709/61/O/N/25 Q3(b))

Your turnIndependent Poisson-approximation check [3 marks]

$X\sim B(1000,0.004)$ . Use a suitable approximation to find $P(X>5)$ .

Show worked answer

X\approx\operatorname{Po}(4)

P(X>5)=1-P(X\le5)

Answer

P(X>5)=0.215

Common mistake

Poisson has one parameter. Write

\operatorname{Po}(np)

, not

\operatorname{Po}(n,p)