Working with lognormal distributions

[PeterVincent]

Hi,

I really would appreciate some help with working with random variables that are log normal distribuited.

Please advise if the following is correct and also any errors with the method even if my way will give the correct numeriacl answer


Have a list of random variables, call this random variable X

the list has a mean = m_x and a variance v_x

if X lognormal then log[x] = y and y is distrib normal with mean m_y, var ss_y

ss_y stands for sigma squared for y, this leads to s_y as sigma y the standard deviatin of y

Y ~ N(m_y, ss_y)

X is lognormal, X ~ LogNormal(m_y, ss_y)


Expected value of X, E[X] = exp(m_y, + 0.5*ss_y)

Variance of X, V[X] = exp(2*m_y + ss_y] * (exp(ss_y) -1)

Setting E[X] = m_x, and V[X] = v_x gives me:

ss_y = variance of Y
m_y = Mean of Y

Now here is the crunch bit

Find the expected proportion of items whose value is less than some value W

Convert W to Y values = log[W]

Probability(X <W) == Prob(log[X] < log[W] == prob(Y < log[W])

= Prob(((Y - m_y)/s_y) < ((log[W] - m_y)/s_y))
= Prob(Z < ((log[W] - m_y)/s_y))
= and get this value from tables, which is the expected proportion.


Many thanks,

PeterVincent

[vinux]

It is correct.