Lesson test: Rare events and the Poisson law

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Rare events and the Poisson law

1.[2p]
What is held fixed while taking the binomial limit that produces the Poisson law?

The product $λ = n p$ , while $n$ grows and $p$ shrinks

Correct

The number of trials $n$

The answer is: The product $\lambda = np$, while $n$ grows and $p$ shrinks

The success probability $p$

The answer is: The product $\lambda = np$, while $n$ grows and $p$ shrinks

The variance $n p (1 - p)$ and the mean separately

The answer is: The product $\lambda = np$, while $n$ grows and $p$ shrinks

2.[1p]
For a Poisson distribution with $λ = 4$ , what is the variance?

CorrectNot quite: 4

3.[2p]
A call centre averages $4$ calls a minute. What is the probability of exactly $2$ calls in a given minute? Give the answer to four decimal places.

CorrectNot quite: 0.1465

4.[2p]
A shop averages $2.5$ customers per five-minute period. What is the probability that a period has none? Give the answer to four decimal places.

CorrectNot quite: 0.0821

5.[2p]
Bortkiewicz recorded $122$ horse-kick deaths over $200$ corps-years. What value of $λ$ does that give?

CorrectNot quite: 0.61

6.[3p]
Clarke's bomb map fitted the Poisson law closely. What did that establish?

The bombs fell at random, and the apparent clusters were what randomness looks like

Correct

The bombs were aimed at particular districts

The answer is: The bombs fell at random, and the apparent clusters were what randomness looks like

The bomb count was too small to analyse

The answer is: The bombs fell at random, and the apparent clusters were what randomness looks like

The squares were the wrong size

The answer is: The bombs fell at random, and the apparent clusters were what randomness looks like

7.[3p]
A set of counts has a sample variance well above its sample mean. What does that suggest?

The events cluster, so the independence the Poisson law assumes fails

Correct

The rate $λ$ has been measured too precisely

The answer is: The events cluster, so the independence the Poisson law assumes fails

The counts follow a Poisson law with a larger $λ$

The answer is: The events cluster, so the independence the Poisson law assumes fails

Nothing: mean and variance are unrelated for count data

The answer is: The events cluster, so the independence the Poisson law assumes fails

8.[2p]
Requests arrive at a server as a Poisson process at $3$ per second. What is the expected number of requests in a half second interval?

CorrectNot quite: 1.5

9.[2p]
Match each dataset to what it counted.

Bortkiewicz 1898
Clarke 1946
Rutherford and Geiger 1910
Weldon 1894

alpha particles per interval
V-1 hits per map square
deaths per cavalry corps per year
fives and sixes per throw of twelve dice

Show the answer

Bortkiewicz 1898: deaths per cavalry corps per year Clarke 1946: V-1 hits per map square Rutherford and Geiger 1910: alpha particles per interval Weldon 1894: fives and sixes per throw of twelve dice

1.[2p] What is held fixed while taking the binomial limit that produces the Poisson law?

2.[1p] For a Poisson distribution with λ=4, what is the variance?

3.[2p] A call centre averages 4 calls a minute. What is the probability of exactly 2 calls in a given minute? Give the answer to four decimal places.

4.[2p] A shop averages 2.5 customers per five-minute period. What is the probability that a period has none? Give the answer to four decimal places.

5.[2p] Bortkiewicz recorded 122 horse-kick deaths over 200 corps-years. What value of λ does that give?

6.[3p] Clarke's bomb map fitted the Poisson law closely. What did that establish?

7.[3p] A set of counts has a sample variance well above its sample mean. What does that suggest?

8.[2p] Requests arrive at a server as a Poisson process at 3 per second. What is the expected number of requests in a half second interval?

9.[2p] Match each dataset to what it counted.

1.[2p]
What is held fixed while taking the binomial limit that produces the Poisson law?

2.[1p]
For a Poisson distribution with $λ = 4$ , what is the variance?

3.[2p]
A call centre averages $4$ calls a minute. What is the probability of exactly $2$ calls in a given minute? Give the answer to four decimal places.

4.[2p]
A shop averages $2.5$ customers per five-minute period. What is the probability that a period has none? Give the answer to four decimal places.

5.[2p]
Bortkiewicz recorded $122$ horse-kick deaths over $200$ corps-years. What value of $λ$ does that give?

6.[3p]
Clarke's bomb map fitted the Poisson law closely. What did that establish?

7.[3p]
A set of counts has a sample variance well above its sample mean. What does that suggest?

8.[2p]
Requests arrive at a server as a Poisson process at $3$ per second. What is the expected number of requests in a half second interval?

9.[2p]
Match each dataset to what it counted.