Mean, variance and Poisson approximation to the binomialEdexcel A-Level Further Maths: Subtopic test

1. 1
   A box contains 40 light bulbs, each independently faulty with probability $0.15$. The number of faulty bulbs in the box is $X$, where $X\sim B(40,0.15)$.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$0.15$
   • B$6$
   • C$34$
   • D$5.1$

   (b)

   Find $\text{Var}(X)$.

   [1 mark]

   • A$6$
   • B$0.1275$
   • C$2.26$
   • D$5.1$

   (c)

   The random variable $Y\sim\text{Po}(\lambda)$ has the same mean as $X$. Write down $\text{Var}(Y)$ and compare it with $\text{Var}(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variable $X\sim B(n,p)$ has mean $12$ and variance $9.6$.

   (a)

   Find the value of $p$.

   [1 mark]

   • A$0.8$
   • B$1.25$
   • C$0.2$
   • D$0.16$

   (b)

   Find the value of $n$.

   [1 mark]

   • A$60$
   • B$48$
   • C$15$
   • D$50$

   (c)

   Use a calculator to find $P(X>15)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory makes pen cartridges. Each cartridge is independently defective with probability $0.004$. Cartridges are packed in boxes of 500 and $X$ is the number of defective cartridges in a box.

   (a)

   Explain why $X$ may be approximated by a Poisson distribution and state the parameter of that distribution.

   [3 marks]

   (b)

   Use the Poisson approximation to find the probability that a box contains at least 4 defective cartridges, and compare it with the exact binomial value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An app developer studies crash rates. App A crashes independently on $2.5\%$ of launches and is launched 120 times in a day; $X$ is the number of crashes. App B crashes independently on $30\%$ of launches and is launched 20 times in a day; $Y$ is the number of crashes.

   (a)

   (i) State the distribution of $X$ and find $E(X)$ and $\text{Var}(X)$.
   (ii) Explain why a Poisson approximation is suitable for $X$, and use it to find $P(X\geq5)$.

   [6 marks]

   (b)

   Explain, with reference to the mean and variance of $Y$, why a Poisson approximation is not suitable for $Y$. Support your answer by finding $P(Y\leq3)$ exactly and using the Poisson approximation.

   [6 marks]

   Total for question 4: 12 marks

End of questions