Bear with me, I cannot format on Stack Exchange, but I will do my best to explain.

I need to calculate the proportion of claims above a certain point.

I have my expectation and variance hats.

I'm being told I should calculate this data from a lognormal distribution above a given point, but to do it via TI 84 I must convert the lognormal parameters and boundaries to normal. I do not know how to do this.

So specifically I have lognormal ML estimators as mu hat = 5.75, sigma squared hat = .16, and I need to know what the proportion that exceed 400 is.

EDIT: I found a calculator online, and I knew I would, but asked this anyway in case I had to calculate the cdf on a test.

The lognormalCDF(400, 5.75, .4) = 0.726965598

So the complement is 1 - 0.726965598 = .273034402 = P(X>400)


Conventionally the $\mu$ and $\sigma$ parameters refer to the mean and standard deviation of the log of the lognormal random variable. If $Y$ is lognormal$(\mu,\sigma^2)$ then $X=\log(Y)$ is normal$(\mu,\sigma^2)$.

Since the logarithm is a monotonic transformation

\begin{eqnarray} P[Y>y]&=&P[\log(Y)>\log(y)]\\&=&P[X>\log(y)]\\&=&P[\frac{X-\mu}{\sigma}>\frac{\log(y)-\mu}{\sigma}]\\&=&P[Z>\frac{\log(y)-\mu}{\sigma}] \end{eqnarray}

which can be looked up in standard normal tables (or evaluated using equivalent functions on a computer).

If you do that correctly on your example you should reproduce the answer you got.

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  • $\begingroup$ So I could calculate a normalcdf((log(400)-5.75)/(.4)), e^99, 0, 1) and that would give me the same answer? I just tried, and received 1. Forgive me I have never met lognormal distributions before now. $\endgroup$ – Anotherreason Oct 3 '17 at 4:38
  • $\begingroup$ what's the $e^{99}$ in there doing? Note that if you have a general normal cdf function you don't need to standardize, you can compare log(400) with a normal$(\mu,\sigma^2)$. Note also that you're finding the probability in the upper tail (above 400), so the normal cdf itself would give you the complement of the tail probability. (Note that there are many discussions of the lognormal on our site - some may have useful insights for you) $\endgroup$ – Glen_b Oct 3 '17 at 4:43

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