Date | May 2014 | Marks available | 21 | Reference code | 14M.2.hl.TZ1.11 |
Level | HL only | Paper | 2 | Time zone | TZ1 |
Command term | Find and Show that | Question number | 11 | Adapted from | N/A |
Question
A random variable X has probability density function
f(x)={ax+b,0,2≤x≤3otherwise,a,b∈R
(a) Show that 5a+2b=2.
Let E(X)=μ.
(b) (i) Show that a=12μ−30.
(ii) Find a similar expression for b in terms of μ.
Let the median of the distribution be 2.3.
(c) (i) Find the value of μ.
(ii) Find the value of the standard deviation of X.
Markscheme
(a) ∫32(ax+b)dx (=1) M1A1
[12ax2+bx]32 (=1) A1
52a+b=1 M1
5a+2b=2 AG
[4 marks]
(b) (i) ∫32(ax2+bx)dx (=μ) M1A1
[13ax3+12bx2]32 (=μ) A1
193a+52b=μ A1
eliminating b M1
eg
193a+52(1−52a)=μ A1
112a+52=μ
a=12μ−30 AG
Note: Elimination of b could be at different stages.
(ii) b=1−52(12μ−30)
=76−30μ A1
Note: This solution may be seen in part (i).
[7 marks]
(c) (i) ∫2.32(ax+b)dx (=0.5) (M1)(A1)
[12ax2+bx]2.32 (=0.5)
0.645a+0.3b (=0.5) (A1)
0.645(12μ−30)+0.3(76−30μ)=0.5 M1
μ=2.34 (=295126) A1
(ii) E(X2)=∫32x2(ax+b)dx (M1)
a=12μ−30=−4021, b=76−30μ=12121 (A1)
E(X2)=∫32x2(−4021x+12121)dx=5.539… (=34963) (A1)
Var(X)=5.539K−(2.341K)2=0.05813… (M1)
σ=0.241 A1
[10 marks]
Total [21 marks]