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

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

1. 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
2. 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
3. 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