Log-normal Distribution - Properties - Partial Expectation

Partial Expectation

The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = EP. For a log-normal random variable the partial expectation is given by

g(k) = \int_k^\infty \!xf(x)\, dx = e^{\mu+\tfrac{1}{2}\sigma^2}\, \Phi\!\left(\frac{\mu+\sigma^2-\ln k}{\sigma}\right).

This formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

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