Two conditions for f(x) to be a pdf?

f(x)\ge0 for all x, and total area \int f(x)\,dx=1.

Define the cumulative distribution function F(x0).

F(x0)=P(X\le x0)=\int{-\infty}^{x0}f(x)\,dx

How do you find an unknown constant k in a pdf?

Set the total area under the pdf equal to 1 and solve.

What is special about the cdf for the second piece of a piecewise pdf?

It must include the probability accumulated in the earlier pieces.

Probability density and cumulative distribution functionsEdexcel International A Level Further Maths: Flashcards

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What is a continuous random variable?

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What is a continuous random variable?: A variable that can take any value in an interval, with probabilities given by areas.
For continuous $X$ , what is $P(X=a)$ ?: $0$
Two conditions for $f(x)$ to be a pdf?: $f(x)\ge0$ for all $x$ , and total area $\int f(x)\,dx=1$ .
How is $P(a<X\le b)$ found from the pdf?: $\int_a^bf(x)\,dx$
Define the cumulative distribution function $F(x_0)$ .: $F(x_0)=P(X\le x_0)=\int_{-\infty}^{x_0}f(x)\,dx$
Write $P(a<X\le b)$ using $F$ .: $F(b)-F(a)$
Write $P(X>a)$ using $F$ .: $1-F(a)$
How do you get $f(x)$ from $F(x)$ ?: $f(x)=\frac{dF(x)}{dx}$
How do you find an unknown constant k in a pdf?: Set the total area under the pdf equal to 1 and solve.
What is special about the cdf for the second piece of a piecewise pdf?: It must include the probability accumulated in the earlier pieces.
$f(x)=\frac{x}{2}$ for $0\le x\le2$ . Find $F(x)$ .: $F(x)=\frac{x^2}{4}$
What are the values of $F$ at the lowest and highest ends of the range?: $0$ and $1$

Exam questions on Probability density and cumulative distribution functions

The continuous random variable $X$ has probability density function $f(x)=kx^2$ for $0\le x\le3$ , and $f(x)=0$ otherwise, where $k$ is a constant.
Find the cumulative distribution function $F(x)$ for $0\le x\le3$ .2 marks
The continuous random variable $Y$ has cumulative distribution function $F(y)=0$ for $y<1$ , $F(y)=\frac{y^2-1}{8}$ for $1\le y\le3$ , and $F(y)=1$ for $y>3$ .
Find $P(1.5<Y\le2.5)$ .2 marks
The continuous random variable $X$ has probability density function $f(x)=kx$ for $0\le x\le2$ , $f(x)=k(6-2x)$ for $2<x\le3$ , and $f(x)=0$ otherwise, where $k$ is a constant.
Show that $k=\frac13$ .3 marks

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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).