Question

Explain how to use a uniform pseudo-random number generator to generate four independent values from distribution for which the p.d.f. is

\(\begin{aligned}g\left( y \right) &= \frac{1}{2}\left( {2y + 1} \right),0 < y < 1,\\ &= 0,otherwise\end{aligned}\)

Short answer

If n independent values \({X_1}, \ldots ,{X_n}\)are produced by the generator, then

the corresponding values \({Y_1}, \ldots ,{Y_n}\) will appear to form a random sample of size n

from the distribution with the c.d.f. G.

Step-by-step solution

Given information

A uniform pseudo-random number generator is to be used to generate four independent values from the distribution for which the p.d.f. y is as follows:

The pdf of the random variable Y is,

\(\begin{aligned}g\left( y \right) &= \frac{1}{2}\left( {2y + 1} \right),0 < y < 1,\\ &= 0,otherwise\end{aligned}\)

Step 2:Obtain the CDF of G from the given pdf

The c.d.f. of G is obtained by integrating the p.d.f. g(y) as follows,

\(\begin{aligned}G\left( y \right) &= \int {g\left( y \right)} dy\\ &= \int\limits_{}^{} {\frac{1}{2}\left( {2y + 1} \right)dy} \\ &= \int\limits_{}^{} {y + \frac{1}{2}dy} \\ &= \frac{{{y^2}}}{2} + \frac{y}{2}\end{aligned}\)

Step 3:Define the Inverse function

Define a random variable x as the pseudo-number that could be generated from the uniform distribution defined over [0,1].,

If a random variable X has the uniform distribution on the interval [0, 1], and if the quantile function \({{\bf{G}}^{{\bf{ - 1}}}}\) is defined, then,

\({\bf{Y = }}{{\bf{G}}^{{\bf{ - 1}}}}\left( {\bf{X}} \right)\).

Hence, if a value of X is produced by a uniform pseudo-random number generator, then the corresponding value of Y will have the desired property.

Thus, for 0 < x <1,

\(\begin{aligned}x &= G\left( y \right)\\x &= \frac{{{y^2}}}{2} + \frac{y}{2} \Rightarrow {y^2} + y - 2x = 0\end{aligned}\)

The roots of the equation could be solved using discriminants for equation \(p{y^2} + qy + r = 0\), are obtained using \(p = 1,q = 1,r = - 2\) as follows,

\(\begin{aligned}y &= \frac{{ - 1 + \sqrt {{1^2} - 4\left( 1 \right)\left( { - 2x} \right)} }}{{2\left( 1 \right)}}\\ &= \frac{{ - 1 + \sqrt {1 + 8x} }}{2}\end{aligned}\)

Thus, the 4 different pseudo numbers are generated using the above equation.

GenerateFour Independent Values from a specified p.d.f.

The next step is to generate four uniform pseudo-random numbers \({x_1},{x_2},{x_3},{x_4}\)

using the generator. Suppose that the four generated values are

\({x_1} = {\rm{0}}{\rm{.4125}},{x_2} = {\rm{0}}{\rm{.0894}},{x_3} = {\rm{0}}{\rm{.8302,}}{{\rm{x}}_4} = 0.6789\)

When these values of \({x_1},{x_2},{x_3}\) are substituted successively into\(\left( 1 \right)\)

the values of y that are obtained are \({y_1} = {\rm{0}}{\rm{.54}},{y_2} = {\rm{0}}{\rm{.15}},{y_3} = {\rm{0}}{\rm{.89,}}{{\rm{y}}_4} = 0.77\). These arethen treated as the observed values of four independent random variables with the distribution for which the p.d.f. is g.

Therefore, for this simulation , \({x_1} = {\rm{0}}{\rm{.4125}},{x_2} = {\rm{0}}{\rm{.0894}},{x_3} = {\rm{0}}{\rm{.8302,}}{{\rm{x}}_4} = 0.6789\) the values obtained are;

\({y_1} = {\rm{0}}{\rm{.54}},{y_2} = {\rm{0}}{\rm{.15}},{y_3} = {\rm{0}}{\rm{.89,}}{{\rm{y}}_4} = 0.77\)