State the cdf of an exponential distribution.

F(x)=1-e^{-\lambda x} for x\ge0.

State two conditions for an exponential model.

Events occur independently (at random) and at a constant average rate.

What does \lambda represent?

The average rate of events per unit time.

Does the pdf increase or decrease with x?

Decreases from f(0)=\lambda: short gaps are most likely.

What must you do if the rate is per hour and the time is in minutes?

Convert the time to hours (or the rate to per minute).

Solve e^{-2\lambda}=0.3 for \lambda.

\lambda=\frac{-\ln0.3}{2}=0.602

Give an example where an exponential model fails.

Breakdowns that cluster after servicing: not independent and not at a constant rate.

Exponential distribution model, pdf and CDFAQA A-Level Further Maths: Flashcards

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

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State the pdf of an exponential distribution with parameter $\lambda$ .: $f(x)=\lambda e^{-\lambda x}$ for $x\ge0$ .
State the cdf of an exponential distribution.: $F(x)=1-e^{-\lambda x}$ for $x\ge0$ .
$\mathrm{P}(X>x)$ for an exponential distribution?: $e^{-\lambda x}$
$\mathrm{P}(a<X<b)$ for an exponential distribution?: $e^{-\lambda a}-e^{-\lambda b}$
State two conditions for an exponential model.: Events occur independently (at random) and at a constant average rate.
What does $\lambda$ represent?: The average rate of events per unit time.
Does the pdf increase or decrease with $x$ ?: Decreases from $f(0)=\lambda$ : short gaps are most likely.
How do you find $F(x)$ from $f(x)$ ?: Integrate $f$ from $0$ to $x$ .
How do you find $f(x)$ from $F(x)$ ?: Differentiate $F(x)$ .
What must you do if the rate is per hour and the time is in minutes?: Convert the time to hours (or the rate to per minute).
Solve $e^{-2\lambda}=0.3$ for $\lambda$ .: $\lambda=\frac{-\ln0.3}{2}=0.602$
Give an example where an exponential model fails.: Breakdowns that cluster after servicing: not independent and not at a constant rate.

Exam questions on Exponential distribution model, pdf and CDF

Calls to a helpline arrive at random, independently of each other, at a constant average rate of 3 per hour. The time, $X$ hours, between successive calls is modelled by an exponential distribution with probability density function $f(x)=3e^{-3x}$ for $x\ge0$ .
Find the probability that the time between two successive calls is between 12 minutes and 30 minutes.2 marks
The lifetime, $T$ years, of a certain type of component is modelled by a continuous random variable with cumulative distribution function $F(t)=1-e^{-0.2t}$ for $t\ge0$ .
The pdf of $T$ is $f(t)=0.2e^{-0.2t}$ for $t\ge0$ . Use integration to find the exact probability that a component fails within 5 years.2 marks
The time, $X$ minutes, between successive arrivals at a supermarket checkout is modelled by an exponential distribution with parameter $\lambda$ , where $\lambda>0$ is a constant. It is known that $\mathrm{P}(X>2)=0.3$ .
Find the value of $\lambda$ .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).