For a log-normal random variable the partial expectation is given by:
2.
Converges in distribution to a quadratic form of independent standard normal random variables.
3.
Are independent and identically distributed normal random variables with expected value zero and variance \ Delta t.
4.
For example, suppose that Y = f ( X ) and X is a normal random variable.
5.
Is satisfied, then a sum of } } converges in distribution to a standard normal random variable, as goes to infinity:
6.
It gives the probability that the value of a standard normal random variable " X " will exceed " x ".
7.
Where z _ { \ alpha / 2 } corresponds to the 1-\ alpha / 2 quantile of a standard normal random variable, and
8.
Where " D " is a diagonal matrix and where " x " represents a vector of uncorrelated standard normal random variables.
9.
For example, a somewhat reasonable model for the distribution of the price of a stock at a fixed future time is the exponential of a normal random variable.
10.
"The radius around the true mean in a bivariate normal random variable, re-written in polar coordinates ( radius and angle ), follows a Hoyt distribution.
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