4.14 Linear combinations of random variables and unbiased estimatesIB Maths: Applications and Interpretation HL: Flashcards

What these 13 flashcards ask

E(aX+b)=?
Var(aX+b)=? (awareness)
Expected value of a linear combination a1X1+\cdots+anXn?
Variance of a1X1+\cdots+anXn for independent variables?
Var(X-Y) for independent X and Y?
Var(X1+X2) versus Var(2X)?
What does "unbiased estimate" mean?
Unbiased estimate of the population mean?
Unbiased estimate of the population variance?
Why is sn^2 not used to estimate \sigma^2?
What is n in n=\sum fi?
Which GDC standard deviation gives s{n-1}?
Variance of the sum of 5 independent copies of R with Var(R)=4?

Exam questions on 4.14 Linear combinations of random variables and unbiased estimates

On Saturdays the number of customers at a café is the random variable $X$ with mean $120$ and variance $64$ . On Sundays the number of customers is the random variable $Y$ with mean $90$ and variance $36$ . $X$ and $Y$ are independent. On a Saturday the café's income is $I=15X+200$ AED.
Find the expected income on a Saturday.2 marks
The masses, in grams, of a random sample of six eggs from a farm are $58, 61, 63, 60, 57, 65$ . Use your GDC where needed.
The variance of the six masses, dividing by $6$ , is $s_n^2=7.56$ . Show how this gives the unbiased estimate $9.07$ from part (b), and explain why this estimate is used rather than $7.56$ .2 marks
A bakery sells a loaf with mass $L$ grams, where $E(L)=800$ and $Var(L)=25$ , and rolls whose individual masses $R$ grams have $E(R)=60$ and $Var(R)=4$ . The mass of the loaf and the masses of different rolls are all independent.
A bag contains one loaf and five rolls, with total mass $W=L+R_1+R_2+R_3+R_4+R_5$ . Find $E(W)$ and $Var(W)$ .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).