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• Marina Defta

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Composition method In this technique , the p.d.f. of the distribution to be simulated, is expressed as a probability mixture of properly selected density functions. Let be of one-parameter density functions, where y is the parameter identifying a unique g(x). If a value of y is drawn from a continuous cumulative function for X will be

and X is sampled from the g(x) for that chosen y, the density function

If f(x) is a probability mass function

where

and are density functions (i = 1, … n), then this technique may be applied for generating a complex distribution from simpler distributions that are themselves easily generated by the inverse transform method or by the acceptance-rejection method. Suppose that we had an efficient method to simulate the value of a random variable having either of the two p.d.f.'s {pj , j>=0} or {qj , j>=0}, and suppose we wanted to simulate the value of a random variable X having p.d.f. P(X = j) = apj + (1 - a)qj

j >= 0,

where 0 < a < 1. One way to simulate such a r.v. X is to that if X1 and X2 are r.v.'s having respective p.d.f. { pj } and { qj }, the r.v. X defined by

note then

will have its mass function. From this it follows that we can generate the value of such a r.v. by first generating a random number U and then generate a value of X1 if U < a and of X2 if U > a. Example 1: Suppose we want to generate the value of an r.v. X such that

By noting that

, where

and we can accomplish this by first generating a random number U and then generating from the discrete uniform over 1, 2, . . . , 10 if U < 0.5 and from the discrete uniform over 6, 7, 8, 9, 10 otherwise. That is, we can simulate X as follows 1) generate a random number U1, 2) generate a random number U2,

3) if U1 < 0.5, set X = j, if or equivalently set X = [10U2] + 1.

, or equivalently if

,

4) if U1 > 0.5, set

or equivalently thus X = [5U2] + 6.

,

If Fi , i =1, . . . , n are distribution functions and ai , i =1, . . . , n are nonnegative numbers summing to 1, then the distribution function F given by

is said to be a composition of the distribution function Fi , i =1, . . . , n . One way to simulate from F is first to simulate a r.v. I, equal to i with probability ai , i =1, . . . , n , and then to simulate from the distribution FI. This approach is often referred to as the composition approach. Example 2: Generate an r.v. from

which can be written

where

Algorithm: 1) generate U1 , U2 from U(0,1).

2)

by inverse transform method

Example 3: Generate an r.v. from

Let

and Thus The first random variable Y in now drawn from distribution determines a particular simply generated from Algorithm: 1) generate

from U(0, 1),

. Once this y is selected, it

The desired random variable X from

is then

2) by inverse transform method 3) if Y is selected then the distribution of X is exponential with parameter Y. Generate X by inverse transform method X = -(1/Y) log U2.