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.

Read more about this topic:  Log-normal Distribution, Properties

Famous quotes containing the words partial and/or expectation:

    “You must not be partial in judging: hear out the small and the great alike; you shall not be intimidated by anyone, for the judgment is God’s.”
    —Bible: Hebrew, Deuteronomy 1:17.

    “In the United States, though power corrupts, the expectation of power paralyzes.”
    —John Kenneth Galbraith (b. 1908)