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Date November 2019 Marks available 4 Reference code 19N.2.AHL.TZ0.H_10
Level Additional Higher Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number H_10 Adapted from N/A

Question

A random variable X has probability density function

f ( x ) = { 3 a , 0 x < 2 a ( x 5 ) ( 1 x ) , 2 x b a b R + 3 < b 5. 0 , otherwise

 

Consider the case where  b = 5 .

Find the value of

Find, in terms of a , the probability that X lies between 1 and 3.

[4]
a.

Sketch the graph of f . State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of a .

[4]
b.

a .

[4]
c.i.

E ( X ) .

[3]
c.ii.

the median of X .

[4]
c.iii.

Markscheme

( P ( 1 < X < 3 ) = ) 1 2 3 a d x + a 2 3 x 2 + 6 x 5 d x        (M1)(A1)(A1)

= 3 a + 11 3 a

= 20 3 a ( = 6.67 a )         A1

[4 marks]

a.

        A4

award A1 for (0, 3 a ), A1 for continuity at (2, 3 a ), A1 for maximum at (3, 4 a ), A1 for (5, 0)

Note: Award A3 if correct four points are not joined by a straight line and a quadratic curve.

[4 marks]

b.

P ( 0 X 5 ) = 6 a + a 2 5 x 2 + 6 x 5 d x        (M1)

= 15 a        (A1)

15 a = 1        (M1)

a = 1 15 ( = 0.0667 )        A1

[4 marks]

c.i.

E ( X ) = 1 5 0 2 x d x + 1 15 2 5 x 3 + 6 x 2 5 x d x        (M1)(A1)

= 2.35       A1

[3 marks]

c.ii.

attempt to use  0 m f ( x ) d x = 0.5        (M1)

0.4 + a 2 m x 2 + 6 x 5 d x = 0.5        (A1)

a 2 m x 2 + 6 x 5 d x = 0.1

attempt to solve integral using GDC and/or analytically       (M1)

1 15 [ 1 3 x 3 + 3 x 2 5 x ] 2 m = 0.1

m = 2.44        A1

[4 marks]

c.iii.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
c.iii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.7—Discrete random variables
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Topic 1—Number and algebra » AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers
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Topic 4—Statistics and probability

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