91Ó°ÊÓ

• Many integrals can be advantageously recast as random variable functions.
• We can estimate integrals that would otherwise be impossible to compute in closed form if we can simulate a large number of random variables with proper distributions.

Random variable \(W\)is defined as

\(W = {\mu _2} + {\sigma _2}{\Phi ^{ - 1}}\left[ {U\Phi \left( {\frac{{{c_2} - {\mu _2}}}{{{\sigma _2}}}} \right)} \right]\)And function \(h\) is defined as

\(h\left( {{x_2}} \right) = \frac{{{{\left( {2\pi \sigma _2^2} \right)}^{ - 1/2}}\exp \left[ {{{\left( {{x_2} - {\mu _2}} \right)}^2}/\left( {2\sigma _2^2} \right)} \right]}}{{\Phi \left[ {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right]}},\;\;\; - \infty < {x_2} \le {c_2}\)

From

\(w = {\mu _2} + {\sigma _2}{\Phi ^{ - 1}}\left[ {u\Phi \left( {\frac{{{c_2} - {\mu _2}}}{{{\sigma _2}}}} \right)} \right]\)

It follows that

\(u = \frac{{\Phi \left[ {\left( {w - {\mu _2}} \right)/{\sigma _2}} \right]}}{{\Phi \left[ {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right]}}\)Is the inverse transformation.

The derivative of \(\Phi \)is the p.d.f. of a standard normal distribution, hence, the derivative of the inverse function is

\(\frac{\partial }{{\partial w}}u = \frac{1}{{\Phi \left[ {\left( {{c_2} - {\mu _2}} \right)/{\sigma _2}} \right]}} \cdot {\left( {2\pi \sigma _2^2} \right)^{ - 1/2}}\exp \left[ {\frac{{{{\left( {{x_2} - {\mu _2}} \right)}^2}}}{{2\sigma _2^2}}} \right] = h\left( {{x_2}} \right)\)

Which is p.d.f. ofW.

This is true because the p.d.f. of random variable with uniform distribution on \([0,1]\) is equal to \(1.\)