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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,0,2x3otherwise,a,bR

(a)     Show that 5a+2b=2.

Let E(X)=μ.

(b)     (i)     Show that a=12μ30.

          (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(152a)=μ     A1

          112a+52=μ

          a=12μ30     AG

 

Note:     Elimination of b could be at different stages.

 

          (ii)     b=152(12μ30)

          =7630μ     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(12μ30)+0.3(7630μ)=0.5     M1

          μ=2.34 (=295126)     A1

          (ii)     E(X2)=32x2(ax+b)dx     (M1)

          a=12μ30=4021, b=7630μ=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]

Examiners report

[N/A]

Syllabus sections

Topic 5 - Core: Statistics and probability » 5.5 » Definition and use of probability density functions.
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