Processing math: 100%

User interface language: English | Español

Date May 2014 Marks available 14 Reference code 14M.3sp.hl.TZ0.3
Level HL only Paper Paper 3 Statistics and probability Time zone TZ0
Command term Determine, Show that, Verify, and Write down Question number 3 Adapted from N/A

Question

(a)     Consider the random variable X for which E(X)=aλ+b, where a and bare constants and λ is a parameter.

Show that Xba is an unbiased estimator for λ.

(b)     The continuous random variable Y has probability density function

f(y)={29(3+yλ),0,forλ3yλotherwise

where λ is a parameter.

          (i)     Verify that f(y) is a probability density function for all values of λ.

          (ii)     Determine E(Y).

          (iii)     Write down an unbiased estimator for λ.

Markscheme

(a)     E(Xba)=aλ+bba     M1A1

=λ     A1

(Therefore Xba is an unbiased estimator for λ)     AG

[3 marks]

 

(b)     (i)     f(y)0     R1

 

Note:     Only award R1 if this statement is made explicitly.

 

          recognition or showing that integral of f is 1 (seen anywhere) R1

          EITHER

          λλ329(3+yλ)dy     M1

          =29[(3λ)y+12y2]λλ3     A1

          =29(λ(3λ)+12λ2(3λ)(λ3)12(λ3)2) or equivalent     A1

          =1

          OR

          the graph of the probability density is a triangle with base length 3 and height 23     M1A1

          its area is therefore 12×3×23     A1

          =1

          (ii)     E(Y)=λλ329y(3+yλ)dy     M1

          =29[(3λ)12y2+13y3]λλ3     A1

          =29((3λ)12(λ2(λ3)2)+13(λ3(λ3)3))     M1

          =λ1     A1A1

 

Note:     Award 3 marks for noting that the mean is 23rds the way along the base and then A1A1 for λ1.

 

Note:     Award A1 for λ and A1 for –1.

 

          (iii)     unbiased estimator: Y+1     A1

 

Note:     Accept ˉY+1.

     Follow through their E(Y) if linear.

 

[11 marks]

 

Total [14 marks]

Examiners report

[N/A]

Syllabus sections

Topic 7 - Option: Statistics and probability » 7.3 » Unbiased estimators and estimates.

View options