# Thread: Difficult Integral standard normal pdf/cdf

1. Originally Posted by BGM
The first four terms vanish due to
The last integral again can be transformed as a bivariate normal integrals

Let X, Y, Z be independent normal random variables with mean 0
and variance respectively

The correlation (covariance)

As a conclusion,
Let

Then

and p approximately equal to 0.290215311627583

The required integral

and approximately equal to 0.864683184411258
Great - thanks

2. BGM: It occurred to me that analytical expressions are possible. That is, given your results and using the standard normal bivariate pdf as you suggested we have for the first integral

and for the second integral

where

and where is the standard normal cdf and and is standard normal pdf.

And, that should do it.

3. Thanks. Learn many things about the normal integration.

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