Part-discrete part-continuous distributionsAQA A-Level Further Maths: Mind map

Exam questions on Part-discrete part-continuous distributions

1. 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
2. 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
3. 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