What is the distribution of aX-bY for independent Normal X,Y?

N\left(a\mux-b\muy,\ a^2\sigmax^2+b^2\sigmay^2\right): variances add.

Why are variances added when variables are subtracted?

Both variables introduce uncertainty, so the spread of the difference is larger, never smaller.

What condition is needed to add variances?

The variables must be independent.

Why does 2X have larger variance than X1+X2?

Because 4\sigma^22\sigma^2: independent values partly cancel each other's variation.

How do you find P(PA) for independent Normal P and A?

Form D=P-A, state its Normal distribution, and find P(D0).

What is the distribution of the mean \bar X of n independent N(\mu,\sigma^2) values?

N\left(\mu,\ \frac{\sigma^2}{n}\right).

What is the first step in a probability question involving several Normal variables?

Define the combined variable and state its distribution N(mean,variance).

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

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If $X\sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ are independent, what is the distribution of $aX+bY$?

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If $X\sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ are independent, what is the distribution of $aX+bY$ ?: $N\left(a\mu_x+b\mu_y,\ a^2\sigma_x^2+b^2\sigma_y^2\right)$ .
What is the distribution of $aX-bY$ for independent Normal $X,Y$ ?: $N\left(a\mu_x-b\mu_y,\ a^2\sigma_x^2+b^2\sigma_y^2\right)$ : variances add.
Why are variances added when variables are subtracted?: Both variables introduce uncertainty, so the spread of the difference is larger, never smaller.
What condition is needed to add variances?: The variables must be independent.
What is the distribution of $2X$ for $X\sim N(\mu,\sigma^2)$ ?: $N(2\mu,\ 4\sigma^2)$ .
What is the distribution of $X_1+X_2$ for independent copies of $X\sim N(\mu,\sigma^2)$ ?: $N(2\mu,\ 2\sigma^2)$ .
Why does $2X$ have larger variance than $X_1+X_2$ ?: Because $4\sigma^2>2\sigma^2$ : independent values partly cancel each other's variation.
How do you find $\mathrm{P}(P>A)$ for independent Normal $P$ and $A$ ?: Form $D=P-A$ , state its Normal distribution, and find $\mathrm{P}(D>0)$ .
What is the distribution of the mean $\bar X$ of $n$ independent $N(\mu,\sigma^2)$ values?: $N\left(\mu,\ \frac{\sigma^2}{n}\right)$ .
Find $\mathrm{Var}(3X-2Y)$ given $\mathrm{Var}(X)=9$ , $\mathrm{Var}(Y)=4$ and independence.: $9(9)+4(4)=97$ .
What distribution does a linear combination of independent Normal variables have?: A Normal distribution.
What is the first step in a probability question involving several Normal variables?: Define the combined variable and state its distribution $N(\text{mean},\text{variance})$ .

Exam questions on Linear combinations of Normal random variables

The mass of an apple, $A$ g, is modelled by $A\sim N(150, 12^2)$ and the mass of a pear, $P$ g, by $P\sim N(200, 15^2)$ . The masses of apples and pears are independent.
Find the probability that a randomly chosen pear is heavier than a randomly chosen apple.2 marks
The independent random variables $X$ and $Y$ are distributed as $X\sim N(20, 3^2)$ and $Y\sim N(15, 2^2)$ . The random variable $W=3X-2Y$ .
Find $\mathrm{P}(W<25)$ .2 marks
The length of a metal rod, $X$ cm, is modelled by $X\sim N(50, 0.4^2)$ . The lengths of different rods are independent.
Two rods are chosen at random and placed end to end. Find the probability that their total length exceeds $100.5$ cm.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).