p(x)=P(X=x), the probability function.

How do you find an unknown constant in a probability function?

Set the sum of all the probabilities equal to 1 and solve.

F(x0)=P(X\le x0)=\sum{x\le x0}p(x)

How do you find P(X=x) from F?

P(X=x)=F(x)-F(x-1) for whole-number values.

How do you find P(X\ge a) from F?

P(X\ge a)=1-F(a-1) for whole-number values.

No; it is non-decreasing from 0 to 1.

Capital X versus lower-case x?

X is the random variable; x is a particular value it can take.

Probability functions and cumulative distribution functionsEdexcel International A Level Maths: Flashcards

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What is a discrete random variable?

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What is a discrete random variable?: A variable that takes separate values, each with a probability.
What does $p(x)$ mean?: $p(x)=P(X=x)$ , the probability function.
Two rules for a probability function?: $0\le p(x)\le1$ and $\sum p(x)=1$ .
How do you find an unknown constant in a probability function?: Set the sum of all the probabilities equal to 1 and solve.
Definition of $F(x_0)$ ?: $F(x_0)=P(X\le x_0)=\sum_{x\le x_0}p(x)$
Value of $F$ at the largest value of $X$ ?: 1
How do you find $P(X=x)$ from $F$ ?: $P(X=x)=F(x)-F(x-1)$ for whole-number values.
How do you find $P(X>a)$ from $F$ ?: $P(X>a)=1-F(a)$
How do you find $P(X\ge a)$ from $F$ ?: $P(X\ge a)=1-F(a-1)$ for whole-number values.
How do you find $P(a<X\le b)$ ?: $F(b)-F(a)$
If $X$ takes whole-number values, what is $F(2.5)$ ?: $F(2.5)=F(2)$
Can F ever decrease?: No; it is non-decreasing from 0 to 1.
Capital X versus lower-case x?: $X$ is the random variable; $x$ is a particular value it can take.

Exam questions on Probability functions and cumulative distribution functions

The discrete random variable $X$ has probability function $P(X=x)=kx$ for $x=1,2,3,4$ , where $k$ is a constant.
Find $P(2\le X<4)$ .2 marks
The discrete random variable $X$ takes the values 1, 2, 3 and 4. Its cumulative distribution function is given by $F(1)=0.15$ , $F(2)=0.40$ , $F(3)=0.75$ and $F(4)=1$ .
Find the probability that $X$ is either 2 or 4.2 marks
The discrete random variable $X$ has probability function $P(X=x)=k(x^2+1)$ for $x=0,1,2,3$ , where $k$ is a constant.
Find the value of $k$ and write down the probability function as a list of probabilities.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).