Part-discrete part-continuous distributionsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

What is a part discrete, part continuous distribution?
What must the point masses and the area under f add up to?
P(X=0)=0.4. What is \int f(x)\,dx over the range?
How do you find an unknown constant in f?
Formula for P(X\le b) when b0?
Is P(X\le0) equal to P(X<0) when P(X=0)0?
Is P(X=a)=0 for every a?
Does P(a<X<b) with a0 include the point mass at 0?
P(X=0)=0.7. What is the median?
Equation for a quartile q with q0?
Is f(a) a probability?
Contribution of a point mass at 0 to E(X)?

Exam questions on Part-discrete part-continuous distributions

The time $X$ minutes that a driver waits at a crossing is modelled as follows. With probability $0.4$ the light is green and $X=0$ . Otherwise $X$ is continuous with probability density function $f(x)=kx$ for $0<x\le2$ , where $k$ is a constant.
Find $P(X>1)$ .2 marks
The daily rainfall $R$ mm at a weather station is modelled as follows. $P(R=0)=0.6$ . On the other days $R$ is continuous with probability density function $f(r)=0.05r$ for $0<r\le4$ .
Find the probability that the rainfall on a given day exceeds $3$ mm.2 marks
A component fails immediately with probability $0.1$ , so its lifetime $X$ years satisfies $P(X=0)=0.1$ . Otherwise $X$ is continuous with probability density function $f(x)=k(6-x)$ for $0<x\le6$ , where $k$ is a constant.
Show that $k=\frac1{20}$ .3 marks

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).