This post is written by Jordan Douglas. If you know some of the quantiles of a probability distribution (eg. P(X < 5) = 0.05, P(X < 10) = 0.95) and you know the name of the distribution but you do not know its parameters, then you can use R to estimate them. This can be done using the cumulative distribution functions (pnorm for normal, plnorm for lognormal, pexp for exponential etc.). Parameter values which yield a small difference between observed and calculated cumulative probabilities can be found using the optim function. For example, suppose that you want to find mu and sigma parameters of a lognormal distribution such that P(X < 5) = 0.05 and P(X < 10) = 0.95. We start at mu = 1 and sigma = 1.

> mu = 1
> sigma = 1
> x1 = 5
> x2 = 10
> p1 = 0.05
> p2 = 0.95
> plnorm(x1, mu, sigma)
[1] 0.7288829
> plnorm(x2, mu, sigma)
[1] 0.9036418

These initial values are a poor fit. Next we generate an error function, in this case we minimize the sum of squared differences between calculated and observed cumulative probabilities.

> errorFn = function(params) (plnorm(x1, params[1], params[2]) - p1)^2 + (plnorm(x2, params[1], params[2]) - p2)^2

Finally, we call optim to find estimates for sigma and mu:

> optim( c(mu, sigma), errorFn)
$par
[1] 1.9560118 0.2107083

$value
[1] 5.499438e-11
...

Thus, the estimates for mu and sigma are 1.956 and 0.211 respectively, which have a sum of squared residuals of 5.5e-11.