Statistics

  1. Apply the Monte Carlo method using 2000 independent realisations of a
    U[0,1] variable to estimate the value of the definite integral
    2 1.96
    2
    1.96
    1 sin( ) 2
    x
    x e dx
    π
    § ·
    ¨ ¸ ¨ ¸ © ¹
    
    ª º
    « » 
    ¬ ¼ ³ .
    Clearly state any changes of variable required and derive the resulting
    equivalent definite integral.
  2. Consider the problem of evaluating the value of the definite integral
    2 2
    0
    cos( ) x e x dx
    f
    
    ³ .
    i) Using the substitution 1 x y e  and 5000 independent
    realisations of a U[0,1] variable, estimate the value of the integral.
    ii) Find an alternative suitable substitution and obtain a second
    estimate of the value of the integral using the same 5000
    realisations of a U[0,1] variable.
    Clearly state your change of variable and derive the resulting
    equivalent definite integral.
    3.
    Consider the problem of evaluating the value of the definite integral
    0
    2
    10
    1
    dx
    x f
    « »
    « » ³ ¬ ¼  .
    i) Using the substitution 2
    1
    1 y
    x 
    and 5000 independent
    realisations of a U[0,1] variable, estimate the value of the integral.
    Hint: Think carefully when calculating x y( )for the required substitution.
    ii) Find an alternative suitable substitution and obtain a second
    estimate of the value of the integral using the same 5000
    realisations of a U[0,1] variable.
    Clearly state your change of variable and derive the resulting
    equivalent definite integral.
    Note: z denotes the floor function which, for any z , takes the
    value of the largest integer z . For example, if z 3.5, then
    2
    10 10 0
    1 ( 3.5) 13.25
    « » « » « » « »

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