What is Var(aX\pm bY) for independent X and Y?

a^2Var(X)+b^2Var(Y), with a plus sign for both sum and difference.

If X\simN(\mux,\sigmax^2) and Y\simN(\muy,\sigmay^2) are independent, what is X+Y?

N(\mux+\muy,\ \sigmax^2+\sigmay^2).

What is X-Y for independent normal X and Y?

N(\mux-\muy,\ \sigmax^2+\sigmay^2): the variances still add.

What is aX+bY for independent normal X and Y?

N(a\mux+b\muy,\ a^2\sigmax^2+b^2\sigmay^2).

How do you find P(XY)?

Find the distribution of X-Y and calculate P(X-Y0).

Why is Var(X1+X2) smaller than Var(2X)?

Independent observations partly cancel, so 2\sigma^2<4\sigma^2. 2X uses the same value twice, so the error doubles.

What do you use to standardise a normal variable?

The standard deviation, \sqrt{variance}: Z=\frac{W-\mu}{\sigma}.

Is a linear combination of independent normal variables normal?

Yes, it is normal with the mean and variance given by the formulae.

What are the steps for a combination problem?

Define the new variable, find its mean and variance, state its normal distribution, then standardise.

Linear combinations of Normal random variablesEdexcel International A Level Further Maths: Flashcards

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What is $\mathrm{E}(aX\pm bY)$?

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What is $\mathrm{E}(aX\pm bY)$ ?: $a\mathrm{E}(X)\pm b\mathrm{E}(Y)$ .
What is $\mathrm{Var}(aX\pm bY)$ for independent $X$ and $Y$ ?: $a^2\mathrm{Var}(X)+b^2\mathrm{Var}(Y)$ , with a plus sign for both sum and difference.
If $X\sim\mathrm{N}(\mu_x,\sigma_x^2)$ and $Y\sim\mathrm{N}(\mu_y,\sigma_y^2)$ are independent, what is $X+Y$ ?: $\mathrm{N}(\mu_x+\mu_y,\ \sigma_x^2+\sigma_y^2)$ .
What is $X-Y$ for independent normal $X$ and $Y$ ?: $\mathrm{N}(\mu_x-\mu_y,\ \sigma_x^2+\sigma_y^2)$ : the variances still add.
What is $aX+bY$ for independent normal $X$ and $Y$ ?: $\mathrm{N}(a\mu_x+b\mu_y,\ a^2\sigma_x^2+b^2\sigma_y^2)$ .
What condition is needed for the variance formula?: $X$ and $Y$ must be independent.
How do you find $\mathrm{P}(X>Y)$ ?: Find the distribution of $X-Y$ and calculate $\mathrm{P}(X-Y>0)$ .
What is the distribution of $X_1+\dots+X_n$ for independent observations from $\mathrm{N}(\mu,\sigma^2)$ ?: $\mathrm{N}(n\mu,\ n\sigma^2)$ .
What is the distribution of $nX$ for $X\sim\mathrm{N}(\mu,\sigma^2)$ ?: $\mathrm{N}(n\mu,\ n^2\sigma^2)$ .
Why is $\mathrm{Var}(X_1+X_2)$ smaller than $\mathrm{Var}(2X)$ ?: Independent observations partly cancel, so $2\sigma^2<4\sigma^2$ . $2X$ uses the same value twice, so the error doubles.
What do you use to standardise a normal variable?: The standard deviation, $\sqrt{\text{variance}}$ : $Z=\frac{W-\mu}{\sigma}$ .
Is a linear combination of independent normal variables normal?: Yes, it is normal with the mean and variance given by the formulae.
What are the steps for a combination problem?: Define the new variable, find its mean and variance, state its normal distribution, then standardise.

Exam questions on Linear combinations of Normal random variables

The time $X$ minutes that a student takes to complete the first section of a test is modelled by $\mathrm{N}(20,4^2)$ . The time $Y$ minutes taken to complete the second section is modelled by $\mathrm{N}(15,3^2)$ . $X$ and $Y$ are independent.
Find the probability that the first section takes longer than the second.2 marks
The random variables $X$ and $Y$ are independent, with $X\sim\mathrm{N}(12,3^2)$ and $Y\sim\mathrm{N}(7,2^2)$ .
Find $\mathrm{P}(3X-2Y>30)$ .2 marks
Four adult passengers, chosen at random, ride in a lift. The mass of an adult passenger is modelled by $\mathrm{N}(75,12^2)$ kg and the mass of the luggage that one passenger carries by $\mathrm{N}(18,5^2)$ kg. All the masses are independent. A calculator may be used.
Find the probability that the total mass of the four passengers, without their luggage, exceeds $320$ kg.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).