How do you get the standard deviation?

Take the square root of the variance.

Var(X); adding a constant does not change the spread.

Can a variance be negative?

No. A negative result means an error.

What makes a game fair?

E(X) equals the entry fee, so the expected net gain is zero.

Must E(X) be a possible value of X?

No; it is a long-run average. The mean score on a fair die is 3.5, which cannot be rolled.

How do you find unknown probabilities a and b?

Use \sum p=1 and a given mean or variance to form simultaneous equations.

Expectation and variance of discrete random variablesEdexcel International A Level Maths: Flashcards

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Formula for $E(X)$?

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Formula for $E(X)$ ?: $E(X)=\sum x\,P(X=x)$
Formula for $E(X^2)$ ?: $E(X^2)=\sum x^2P(X=x)$
Formula for $\mathrm{Var}(X)$ ?: $\mathrm{Var}(X)=E(X^2)-[E(X)]^2$
How do you get the standard deviation?: Take the square root of the variance.
Formula for $E(aX+b)$ ?: $aE(X)+b$
Formula for $\mathrm{Var}(aX+b)$ ?: $a^2\mathrm{Var}(X)$
What is $\mathrm{Var}(X+5)$ ?: $\mathrm{Var}(X)$ ; adding a constant does not change the spread.
What is $\mathrm{Var}(-X)$ ?: $\mathrm{Var}(X)$ , because $(-1)^2=1$ .
Can a variance be negative?: No. A negative result means an error.
If $\mathrm{Var}(X)$ and $E(X)$ are known, find $E(X^2)$ .: $E(X^2)=\mathrm{Var}(X)+[E(X)]^2$
What makes a game fair?: $E(X)$ equals the entry fee, so the expected net gain is zero.
Must E(X) be a possible value of X?: No; it is a long-run average. The mean score on a fair die is 3.5, which cannot be rolled.
How do you find unknown probabilities a and b?: Use $\sum p=1$ and a given mean or variance to form simultaneous equations.

Exam questions on Expectation and variance of discrete random variables

The discrete random variable $X$ has probability distribution $P(X=0)=0.2$ , $P(X=1)=0.3$ , $P(X=2)=0.4$ and $P(X=3)=0.1$ .
Find $E(3X+2)$ .2 marks
The random variable $W$ has $E(W)=10$ and $\mathrm{Var}(W)=6$ . The random variable $V$ is defined by $V=3W-4$ .
Find $E(W^2)$ .2 marks
The discrete random variable $X$ takes the values 1, 2, 3 and 4 with $P(X=1)=0.1$ , $P(X=2)=a$ , $P(X=3)=0.4$ and $P(X=4)=b$ , where $a$ and $b$ are constants. It is given that $E(X)=2.9$ .
Find the values of $a$ and $b$ .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).