Editing Log-normal distribution (section)

=== Partial expectation ===

The partial expectation of a random variable <math>X</math> with respect to a threshold <math>k</math> is defined as

<math display="block"> g(k) = \int_k^\infty x \, f_X(x \mid X > k)\, dx . </math>

Alternatively, by using the definition of [[conditional expectation]], it can be written as <math>g(k) = \operatorname{E}[X\mid X>k] \Pr(X>k)</math>. For a log-normal random variable, the partial expectation is given by:

<math display="block">\begin{align}
g(k) &= \int_k^\infty x f_X(x \mid X > k)\, dx \\[1ex]
&= e^{\mu+\tfrac{1}{2} \sigma^2}\, \Phi{\left(\frac{\mu-\ln k}{\sigma} - \sigma\right)}
\end{align} </math>

where <math>\Phi</math> is the [[normal cumulative distribution function]]. The derivation of the formula is provided in the [[Talk:Log-normal distribution|Talk page]]. The partial expectation formula has applications in [[insurance]] and [[economics]], it is used in solving the partial differential equation leading to the [[Black–Scholes formula]].