Cumulative distribution functionsAQA A-Level Further Maths: Mind map

Exam questions on Cumulative distribution functions

1. The continuous random variable $X$ has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=\frac{3x^2-x^3}{4}$ for $0\le x\le2$, and $F(x)=1$ for $x>2$.
   Find the probability density function $f(x)$ of $X$.2 marks
2. The continuous random variable $X$ has probability density function $f(x)=\frac{3}{x^4}$ for $x\ge1$, and $f(x)=0$ otherwise.
   Find the median of $X$.2 marks
3. The continuous random variable $X$ has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=k(x^2+2x)$ for $0\le x\le2$, and $F(x)=1$ for $x>2$, where $k$ is a constant.
   Find the value of $k$.3 marks