How do you find an unknown constant in a distribution?

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

How do you find the standard deviation?

Take the square root of the variance.

What is the mode of a DRV?

The value with the greatest probability.

How do you find the median of a DRV?

The smallest m with P(X\le m)\ge0.5.

Can a variance be negative?

No; if you get one, recheck E(X^2).

Distributions and expectation of DRVsAQA A-Level Further Maths: Flashcards

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What two rules must a probability distribution satisfy?

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What two rules must a probability distribution satisfy?: Each $0\le p\le1$ and $\sum p=1$ .
How do you find an unknown constant in a distribution?: Set the sum of all probabilities equal to $1$ and solve.
Formula for $E(X)$ ?: $E(X)=\sum x_ip_i$ .
Formula for $E(X^2)$ ?: $E(X^2)=\sum x_i^2p_i$ .
Formula for $\mathrm{Var}(X)$ ?: $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$ .
How do you find the standard deviation?: Take the square root of the variance.
What is the mode of a DRV?: The value with the greatest probability.
How do you find the median of a DRV?: The smallest $m$ with $P(X\le m)\ge0.5$ .
Is $E(X^2)=[E(X)]^2$ ?: No; they differ by $\mathrm{Var}(X)$ .
Can the mean of a DRV be a value that $X$ cannot take?: Yes; it is a long-run average.
Can a variance be negative?: No; if you get one, recheck $E(X^2)$ .
$P(X\ge3)$ for $X=1,2,3,4$ in terms of probabilities?: $P(X=3)+P(X=4)$ .

Exam questions on Distributions and expectation of DRVs

The discrete random variable $X$ has probability distribution $P(X=x)=kx$ for $x=1,2,3,4$ , and $P(X=x)=0$ otherwise, where $k$ is a constant.
Find $E(X)$ .2 marks
The discrete random variable $X$ has $P(X=1)=0.1$ , $P(X=2)=0.3$ , $P(X=3)=0.4$ and $P(X=4)=0.2$ .
State the mode of $X$ and find the median of $X$ .2 marks
The discrete random variable $X$ takes the values $1,2,3,4$ with $P(X=1)=0.2$ , $P(X=2)=p$ , $P(X=3)=q$ and $P(X=4)=0.3$ , where $p$ and $q$ are constants. It is given that $E(X)=2.6$ .
Find the values of $p$ and $q$ .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).