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Simulation Methods

(a) Monte Carlo Technique

(i) Generate random numbers µ from (0, 1)

(ii) Generate random observations from any desired probability distribution.


(b) Method of Inversion

(i) Generate a random number µ from (0, 1)

(ii) Find the distribution function F(x) of the random variable X

(iii)    Set                   F(x i) = u i

i  = f -1(u i)

which gives the random observation from the probability distribution of X.


(c) Rejection Method

Let the pdf f(x) over a finite interval [A, B] and C be the mode of the distribution.

(i) Generate two random numbers u1 and u2 from (0, 1).


(ii) Take a point on x-axis as

p = A+ (B- A) u1


(iii) Take a point on the vertical axis as

q = c.u2


(iv) If q ≤ f (p) accept the pair (p, q) else reject µ pair. All the rejected pairs lie outside the boundary of the curve j(x).