How do you get the standard deviation?

Take the positive square root of the variance.

Why does b not appear in Var(aX+b)?

Adding a constant shifts all values equally, so the spread is unchanged.

No. E(X^2)-[E(X)]^2=Var(X)\ge0.

Can a variance be negative?

No. A negative answer means an arithmetic error.

How do you find Var(6X^{-1})?

36\,Var(X^{-1})=36\left(E(X^{-2})-[E(X^{-1})]^2\right).

Mean and variance of a CRVAQA A-Level Further Maths: Flashcards

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Formula for $E(X)$ of a continuous random variable?

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Formula for $E(X)$ of a continuous random variable?: $E(X)=\int xf(x)\,dx$ over the range of $X$ .
Formula for $E(X^2)$ ?: $E(X^2)=\int x^2f(x)\,dx$
Formula for $\text{Var}(X)$ ?: $\text{Var}(X)=E(X^2)-[E(X)]^2$
How do you get the standard deviation?: Take the positive square root of the variance.
$E(aX+b)$ ?: $aE(X)+b$
$\text{Var}(aX+b)$ ?: $a^2\,\text{Var}(X)$
Why does $b$ not appear in $\text{Var}(aX+b)$ ?: Adding a constant shifts all values equally, so the spread is unchanged.
Formula for $E(g(X))$ ?: $E(g(X))=\int g(x)f(x)\,dx$
Is $E(X^2)=[E(X)]^2$ ?: No. $E(X^2)-[E(X)]^2=\text{Var}(X)\ge0$ .
Can a variance be negative?: No. A negative answer means an arithmetic error.
$E(X)=5$ and $\text{Var}(X)=4$ . Find $\text{Var}(2-3X)$ .: $9\times4=36$
How do you find $\text{Var}(6X^{-1})$ ?: $36\,\text{Var}(X^{-1})=36\left(E(X^{-2})-[E(X^{-1})]^2\right)$ .
What is the mean of a symmetric pdf?: The line of symmetry.

Exam questions on Mean and variance of a CRV

The continuous random variable $X$ has probability density function $f(x)=\frac{3x^2}{8}$ for $0\le x\le2$ , and $f(x)=0$ otherwise.
Find $\text{Var}(X)$ .2 marks
The random variable $X$ has mean $E(X)=5$ and variance $\text{Var}(X)=4$ .
The random variable $Y=2-3X$ . Find $E(Y)$ and $\text{Var}(Y)$ .2 marks
The continuous random variable $X$ has probability density function $f(x)=\frac{2}{x^2}$ for $1\le x\le2$ , and $f(x)=0$ otherwise.
Find $E(X)$ .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).