Poisson and binomial distributions questions

Questions to attempt

6 questions on Poisson and binomial distributions, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. FS1-2.1 · The Poisson distribution. The additive property of Poisson distributions.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

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

   (3)

   (Total for Question 1 is 3 marks)

   Reveal answer and method

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2. FS1-2.1 · The Poisson distribution. The additive property of Poisson distributions.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   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.

   (2)

   (Total for Question 2 is 2 marks)

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3. FS1-2.2 · The mean and variance of the binomial distribution and the Poisson distribution.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

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

   (4)

   (Total for Question 3 is 4 marks)

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4. FS1-2.2 · The mean and variance of the binomial distribution and the Poisson distribution.

   Q4

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   4.

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

   (2)

   (Total for Question 4 is 2 marks)

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5. FS1-2.3 · The use of the Poisson distribution as an approximation to the binomial distribution.

   Q5

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   5.

   The number of flawed seals in a batch has distribution $X\sim\operatorname{Bin}(400,0.01)$. Use a Poisson approximation to estimate $P(X\geq6)$.

   (4)

   (Total for Question 5 is 4 marks)

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6. FS1-2.3 · The use of the Poisson distribution as an approximation to the binomial distribution.

   Q6

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   6.

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

   (3)

   (Total for Question 6 is 3 marks)

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FS1-2 Poisson and binomial distributions revision notes