Date | May 2021 | Marks available | 2 | Reference code | 21M.2.AHL.TZ2.6 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Show that | Question number | 6 | Adapted from | N/A |
Question
A continuous random variable X has the probability density function fn given by
fn(x)={(n+1)xn, 0, 0≤x≤1otherwise
where n∈ℝ, n≥0.
Show that E(X)=n+1n+2.
Show that Var(X)=n+1(n+2)2(n+3).
Markscheme
E(X)=(n+1)1∫0xn+1dx M1
=(n+1)[xn+2n+2]10 A1
leading to E(X)=n+1n+2 AG
[2 marks]
METHOD 1
use of Var(X)=E(X2)-[E(X)]2 M1
Var(X)=(n+1)1∫0xn+2dx-(n+1n+2)2
=(n+1)[1n+3xn+3]10-(n+1n+2)2
=n+1n+3-(n+1n+2)2 A1
=(n+1)(n+2)2-(n+1)2(n+3)(n+2)2(n+3) M1
EITHER
=(n+1)(n2+4n+4-(n2+4n+3))(n+2)2(n+3) A1
OR
=(n3+5n2+8n+4)-(n3+5n2+7n+3)(n+2)2(n+3) A1
THEN
so Var(X)=n+1(n+2)2(n+3) AG
METHOD 2
use of Var(X)=E(X-E(X))2 M1
Var(X)=(n+1)1∫0(x-n+1n+2)2xndx
=(n+1)[1n+3xn+3-2(n+1)(n+2)2xn+2+n+1(n+2)2xn+1]10
=n+1n+3-(n+1n+2)2 A1
=(n+1)((n+2)2-(n+1)(n+3))(n+2)2(n+3) M1
EITHER
=(n+1)(n2+4n+4-(n2+4n+3))(n+2)2(n+3) A1
OR
=(n3+5n2+8n+4)-(n3+5n2+7n+3)(n+2)2(n+3) A1
THEN
so Var(X)=n+1(n+2)2(n+3) AG
[6 marks]