Probability density and cumulative distribution functionsEdexcel A-Level Further Maths: Mind map

Exam questions on Probability density and cumulative distribution functions

1. The continuous random variable $X$ has probability density function $\mathrm{f}(x)=kx^2$ for $0\leq x\leq3$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.
   Find the cumulative distribution function $\mathrm{F}(x)$ for $0\leq x\leq3$.2 marks
2. The continuous random variable $X$ has cumulative distribution function $\mathrm{F}(x)=0$ for $x<1$, $\mathrm{F}(x)=\dfrac{x^2-1}{8}$ for $1\leq x\leq3$, and $\mathrm{F}(x)=1$ for $x>3$.
   Find $\mathrm{P}(1.5<X\leq2.5)$.2 marks
3. The time $X$, in hours, that a customer waits for a delivery has probability density function $\mathrm{f}(x)=kx$ for $0\leq x\leq2$, $\mathrm{f}(x)=2k$ for $2<x\leq4$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.
   Show that $k=\frac16$.3 marks