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Assume I have two values which represent two quantiles for the same lognormal and/or normal distribution.

How can I determine the parameters of the distribution? 

EG, 
F(5,000) = 0.9   # P(X < 5,000) = .90
F(1,500) = 0.75  # P(X < 1,500) = .75
Glen_b
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GPB
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    Do you need to know about the normal distribution (e.g. the title of your question) or the lognormal distribution (in the text)? $\Phi(\cdot)$ usually denotes the standard normal distribution function... – Dilip Sarwate Dec 27 '15 at 19:16
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    Furthermore... the hypothesis seems to be that F(5.0)=0.9 and F(1.5)=0.5 where F(x)=P(X<x), not what is written in the question. – Did Dec 27 '15 at 19:30

1 Answers1

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Firstly, if you're dealing with lognormal quantiles (values exceeding some proportion $p$ of the population distribution), take logs to convert them to normal quantiles.

The parameters of the lognormal are parameters of the normal you get after taking logs, so we have now reduced it to the problem of identifying the parameters of a normal given two quantiles.

[I assume these are known population values rather than estimated values (such as from a sample).]

Let $x_p$ and $x_q$ be the two quantiles.

Then you have two equations in two unknowns, each of the form:

$\Phi(\frac{x_p-\mu}{\sigma})=p$

We can rewrite this as a linear equation in $\mu$ and $\sigma$:

$x_p=\mu\,+\,\Phi^{-1}(p)\,\sigma$

Explicitly evaluate $\Phi^{-1}(p)$ and $\Phi^{-1}(q)$ -- that is find the numerical value of the p-quantile and q-quantile of a standard normal. e.g. if $p=0.9$, $\Phi^{-1}(p)=1.28155$.

So you have two linear equations in two unknowns -- a very standard problem; it's easy to eliminate $\mu$, leaving an equation in $\sigma$ which is readily solved for $\sigma$. You can then substitute back to solve for $\mu$.

Glen_b
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