Sums of independent random variablesAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

E(X+Y) for any random variables?
Var(X+Y) for independent X and Y?
Does E(X+Y)=E(X)+E(Y) need independence?
Does Var(X+Y)=Var(X)+Var(Y) need independence?
Are standard deviations added for a sum of independent variables?
Var(X)=9, Var(Y)=16, independent. Standard deviation of X+Y?
Does the sum rule apply when one variable is discrete and the other continuous?
X1 and X2 are independent copies of X. Var(X1+X2)?
Var(2X) in terms of Var(X)?
Why is Var(X1+X2)\neVar(2X)?
What must you state before adding variances?
Formula for the variance of a single variable from its pdf?

Exam questions on Sums of independent random variables

$X$ and $Y$ are independent random variables with $E(X)=12$ , $\text{Var}(X)=9$ , $E(Y)=7$ and $\text{Var}(Y)=16$ .
A student claims that the standard deviation of $X+Y$ is $3+4=7$ . Show that this is wrong and state the correct value.2 marks
The times, in minutes, taken to prepare and to cook a dish are modelled by independent random variables $X$ and $Y$ , with $E(X)=8.5$ , $\text{Var}(X)=1.44$ , $E(Y)=12.3$ and $\text{Var}(Y)=2.56$ . The total time is $T=X+Y$ .
Find the standard deviation of $T$ , and state the assumption that makes it valid to add the variances.2 marks
The continuous random variable $X$ has probability density function $f(x)=\frac{x}{2}$ for $0\le x\le2$ , and $f(x)=0$ otherwise. The discrete random variable $Y$ is the score when a fair six-sided die is rolled, so $E(Y)=\frac72$ and $\text{Var}(Y)=\frac{35}{12}$ . $X$ and $Y$ are independent.
Find $E(X+Y)$ .3 marks

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).