Question

To evaluate the system parameters, one uses Monte Carlo methodology and simulates one of its vital characteristics X, a continuous random variable with the density

f(x) = 2/9 (1+x) if -1 < x < 2, otherwise 0

Explain how one can generate X, given a computer-generated Standard Uniform variable U. If U = 0.2396, compute X.

Answer 1

(a)

Given: Probability Density Function of X:

,

                        for - 1 < x < 2

      =0, otherwise

The Cumulative Distribution Function F(x) is given by:

between limits - 1 to x.

Applying limits, we get:

Given: a computer-generated Standard Uniform variable U.

Since the Cumulative Distribution Function varies from0 to 1, itcan be equared to U as follows:

So,

we get:

So, we get:

i.e.,

Taking positive sign, we get:

One can genrate X by the above equation.

(b)

For U = 0.2396, we get:

So,

Answer is:

0.4684