What is the formula for Var(X)?

Var(X)=E(X^2)-\mu^2=\sum x^2P(X=x)-\mu^2

Is E(1/X) equal to 1/E(X)?

No. Work out \sum\frac1xP(X=x).

What condition do the probabilities of a discrete random variable satisfy?

\sum P(X=x)=1, with each probability between 0 and 1.

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

Set the sum of the probabilities equal to 1 and solve for k.

A model has mean 1.6 and variance 0.84, but the data show mean 1.9 and variance 1.69. What do you conclude?

The model underestimates the mean and the spread, so it is not suitable.

Why is E(X^2)\geq[E(X)]^2?

Their difference is the variance, which cannot be negative.

Mean and variance of discrete distributionsEdexcel A-Level Further Maths: Flashcards

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What is the formula for $E(X)$?

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What is the formula for $E(X)$ ?: $E(X)=\sum xP(X=x)$
What is the formula for $\text{Var}(X)$ ?: $\text{Var}(X)=E(X^2)-\mu^2=\sum x^2P(X=x)-\mu^2$
How do you find $E(X^2)$ ?: $\sum x^2P(X=x)$
What is the formula for $E(g(X))$ ?: $\sum g(x)P(X=x)$
Is $E(1/X)$ equal to $1/E(X)$ ?: No. Work out $\sum\frac1xP(X=x)$ .
What is $E(aX+b)$ ?: $aE(X)+b$
What condition do the probabilities of a discrete random variable satisfy?: $\sum P(X=x)=1$ , with each probability between $0$ and $1$ .
How do you find an unknown constant $k$ in a probability function?: Set the sum of the probabilities equal to $1$ and solve for $k$ .
What is the standard deviation in terms of the variance?: $\sigma=\sqrt{\text{Var}(X)}$
What is a fair game?: A game with expected profit $0$ .
A model has mean 1.6 and variance 0.84, but the data show mean 1.9 and variance 1.69. What do you conclude?: The model underestimates the mean and the spread, so it is not suitable.
Why is $E(X^2)\geq[E(X)]^2$ ?: Their difference is the variance, which cannot be negative.

Exam questions on Mean and variance of discrete distributions

The discrete random variable $X$ has $P(X=1)=0.1$ , $P(X=2)=0.3$ , $P(X=3)=0.4$ and $P(X=4)=0.2$ .
Find $E\left(\frac1X\right)$ .2 marks
The discrete random variable $Y$ has probability function $P(Y=y)=ky$ for $y=1,2,3,4,5$ , where $k$ is a constant.
Find $\text{Var}(Y)$ .2 marks
A fairground game costs £3 to play. A player is paid £20 with probability $0.05$ , £6 with probability $0.2$ and nothing otherwise. Let $X$ be the player's profit in pounds, where profit is the payout minus the £3 cost.
Find $E(X)$ and interpret your answer in context.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).