Mean, variance and Poisson approximation to the binomialEdexcel A-Level Further Maths: Mind map

Exam questions on Mean, variance and Poisson approximation to the binomial

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 $X\sim B(40,0.15)$.
   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
2. The random variable $X\sim B(n,p)$ has mean $12$ and variance $9.6$.
   Use a calculator to find $P(X>15)$.2 marks
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.
   Explain why $X$ may be approximated by a Poisson distribution and state the parameter of that distribution.3 marks