11 Turn over Question 4 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 4 is 7 marks) 12 5 y x O 2π Figure 1 The random variable X has probability density function f (x) and Figure 1 shows a sketch of f (x) where f (x) = (1 cos ) 0 2 0 otherwise k x x π −    (a) Show that k = 1 2π (3) The random variable Y ~ N ( μ, σ 2) and E(Y ) = E(X ) The probability density function of Y is g ( y), where g ( y) = 2 1 2 1 e 2 y μ σ σ π −   −               − ∞ < y < ∞ Given that g ( μ) = f ( μ) (b) find the exact value of σ (3) (c) Calculate the error in using P 3 2 2 π π Y       < < as an approximation to P 3 2 2 π π X       < < (4) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________