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Practice — Statistics bridge for Further Mathematics • 0.8 Poisson distribution

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1. Assessing a Poisson model
Difficulty: ★☆☆
2164

Which description satisfies the usual constant-rate Poisson-process assumptions for visitors arriving at an information desk?

Hint
Look for independent random arrivals, a constant rate and no systematic batches.
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2. A zero-count probability
Difficulty: ★☆☆
2165

\(X\sim\mathrm{Po}(1.8)\). Find \(P(X=0)\), to 4 d.p.

Hint
Substitute the count zero into the Poisson formula and simplify before evaluating.
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3. Mean and standard deviation
Difficulty: ★☆☆
2166

\(X\sim\mathrm{Po}(6.25)\). Which pair gives the mean and standard deviation of \(X\)?

Hint
A Poisson mean equals its variance, but not generally its standard deviation.
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4. Scaling the time interval
Difficulty: ★★☆
2167

Emails arrive according to a Poisson process at a constant rate of 9 per hour. Find the probability of exactly two emails in 20 minutes, to 4 d.p.

Hint
Convert 20 minutes to hours before choosing the Poisson parameter.
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5. At most one event
Difficulty: ★★☆
2168

The number of surface flaws on a panel has model \(X\sim\mathrm{Po}(2.6)\). Find \(P(X\le1)\), to 4 d.p.

Hint
At most one includes two possible count values.
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6. An upper tail
Difficulty: ★★☆
2169

\(X\sim\mathrm{Po}(5)\). Find \(P(X\ge2)\), to 4 d.p.

Hint
Write the complement as a short sum over the excluded integer counts.
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7. No events in a scaled interval
Difficulty: ★★☆
2170

Particles are detected according to a Poisson process at a constant mean rate of \(0.8\) per minute. Find the probability of no detections in \(2.5\) minutes, to 4 d.p.

Hint
The expected count grows in direct proportion to the interval length.
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8. Expected category frequency
Difficulty: ★★☆
2171

For each of 150 independent observation intervals, the count has a Poisson distribution with mean \(1.2\). Find the expected number of intervals with two or more events, to 1 d.p.

Hint
Calculate the probability of the category first, then multiply by the number of intervals.
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9. Recovering the effect of a longer interval
Difficulty: ★★★
2172

Calls follow a Poisson process with a constant rate. The probability of no calls in 30 minutes is \(0.30\). Find the probability of at least one call in one hour.

Hint
Split the hour into two disjoint half-hour intervals and use independence for no calls.
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10. Recovering a parameter from adjacent probabilities
Difficulty: ★★★
2173

\(X\sim\mathrm{Po}(\lambda)\), where \(\lambda>0\). It is known that \(P(X=2)=3P(X=1)\). Find \(P(X=0)\), to 4 d.p.

Hint
Substitute both Poisson formulas and cancel their common positive factors.
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