Mean and variance, and link to Poisson processesAQA A-Level Further Maths: Mind map

Exam questions on Mean and variance, and link to Poisson processes

1. The lifetime, $X$ days, of a type of battery is modelled by an exponential distribution with parameter $\lambda=0.05$.
   Find the probability that a battery lasts longer than its mean lifetime.2 marks
2. Particles strike a detector at random, independently of each other, at a constant average rate of 12 per hour, so the number of strikes follows a Poisson process. Let $T$ minutes be the time between two successive strikes.
   Find the standard deviation of $T$ in minutes.2 marks
3. The continuous random variable $X$ has an exponential distribution with parameter $\lambda>0$, so that $f(x)=\lambda e^{-\lambda x}$ for $x\ge0$. You may use the fact that $xe^{-\lambda x}\to0$ and $x^2e^{-\lambda x}\to0$ as $x\to\infty$.
   Prove that $\mathrm{E}(X)=\frac{1}{\lambda}$.3 marks