Suppose X is a continuous random variable on [0,∞) with
probability density function,where α > 0 and β > 0 are
parameters.
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a Brº-1 2. Suppose X is a continuous random variable on (0,00) with probability density func- tion g(x) >0. (3+ca)? where a > 0 and ß > 0 are parameters. = (a) Specify an algorithm to simulate from g using the inverse-transform method. [3 marks] (b) Implement the algorithm from part(a) in Python or R and provide the code. Simulate 105 random variables from g 2 and ß = 3. Provide a plot of the histogram together with the probability density function. [2 marks] (c) We want to construct an algorithm to simulate from the distribution with prob- ability density function with a x-1/2 f(x) = = == X>0 VTI)(1+x)2
where I is the gamma function T(6) = 1926-= xb-1e-2 dx. е = We decide to use the acceptance-rejection with the distribution g as the proposal distribution setting a = 1/2 and B = 1/V3. i. Show that f/g attains a maximum at r* = 1/3. [2 marks] ii. Specify an algorithm to simulate from f using the acceptance-rejection method. [2 marks] iii. Determine the expected number of random variables from g that need to be generated to produce a single random variable from f? The answer can involve the gamma function.) [1 mark] (d) Implement the algorithm from part (c) in Python or R and provide the code. Simulate 105 random variables from f. Provide a plot of the histogram together with the probability density function. [2 marks] a
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