Cumulative distribution functionsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

Definition of F(x)?
How do you get f from F?
How do you get F from f?
F(x) below the range and above the range?
P(a<X<b) in terms of F?
P(Xa) in terms of F?
Equation for the median using F?
Equations for the quartiles using F?
Is a cdf ever decreasing?
What is f(x) outside the range?
How do you find an unknown constant in a cdf?
A cdf has pieces. How do you pick the piece to solve F=\frac12 in?

Exam questions on Cumulative distribution functions

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
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
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

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).