State the standard deviation of an exponential distribution.

\frac{1}{\lambda}, equal to the mean.

Why does the boundary term vanish in the proof?

xe^{-\lambda x}\to0 as x\to\infty, and it is 0 at x=0.

What distribution has the time between events of a Poisson process?

Exponential with parameter \lambda, the rate of the process.

What is the mean gap if events occur 12 times per hour?

\frac{1}{12} hour, which is 5 minutes.

Why is P(gapt)=e^{-\lambda t}?

It is the Poisson probability of zero events in time t.

When does the Poisson process model fail?

When events are not independent, or the rate is not constant.

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

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State the mean of an exponential distribution with parameter $\lambda$.

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State the mean of an exponential distribution with parameter $\lambda$ .: $\frac{1}{\lambda}$
State the variance of an exponential distribution.: $\frac{1}{\lambda^2}$
State the standard deviation of an exponential distribution.: $\frac{1}{\lambda}$ , equal to the mean.
State $\mathrm{E}(X^2)$ for an exponential distribution.: $\frac{2}{\lambda^2}$
Which integration technique proves the mean?: Integration by parts with $u=x$ .
Why does the boundary term vanish in the proof?: $xe^{-\lambda x}\to0$ as $x\to\infty$ , and it is $0$ at $x=0$ .
What distribution has the time between events of a Poisson process?: Exponential with parameter $\lambda$ , the rate of the process.
Number of events in time $t$ for a Poisson process of rate $\lambda$ ?: Poisson with mean $\lambda t$ .
What is the mean gap if events occur 12 times per hour?: $\frac{1}{12}$ hour, which is 5 minutes.
Why is $\mathrm{P}(\text{gap}>t)=e^{-\lambda t}$ ?: It is the Poisson probability of zero events in time $t$ .
What does a higher $\lambda$ do to the mean gap?: It reduces it.
When does the Poisson process model fail?: When events are not independent, or the rate is not constant.

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

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).