find the σ and the μ of the lognormal function, having 1st percentile and the 99th percentile

I have to build find the σ and the μ of the lognormal function, having the value of the 1st percentile (which is 30) and the 99th percentile (which is 60).

Do you have any suggestion to do it simply with matlab

Thank you in advance (be nice, I am not a statistic guru, I have already tried to solve it by myself searching a lot in statistic book)

[Adam Danz]

"I have to build find the σ and the μ of the lognormal function"

So far the problem is easy. Assuming these variables carry the conventional meanings, μ is just the function mean() and σ is just the function std()

"...having the value of the 1st percentile (which is 30) and the 99th percentile (which is 60)."

This is where I start to lose you. Are you trying to create the lognormal distribution with those percentile requirements or do you have those data already and are merely trying to compute σ and μ ?

[SYML2nd]

I am trying to create the lognormal distribution with those percentile requirements

Answer 1

[the cyclist]

First you need to understand the math problem, then you can use MATLAB to solve numerically.

You can find the CDF of the lognormal distribution in this wikipedia section. So, you'll have two simultaneous equations (each equation is a function of σ and μ):

• Eq 1: cdf of the lognormal at 30 = 0.01
• Eq 2: cdf of the lognormal at 60 = 0.99

This is two equations in two unknowns, and with bit of manipulation, I believe can make this two linear equations in two unknowns.

Since this seems like homework, I think maybe that's enough of a hint to get you going.

[SYML2nd]

It is not an homework. Since the cdf is 0.5+erf((log(x)-mu)/(sqrt(2)*sigma)) I have tried to do in this way (see the attached code), but when I do the reverse, trying to find the percentile 0.01 from the built curve the result is not 30. What I did wrongly in your opinion?

a=30
b=60
syms mu sigma 

[solx,soly] = solve( 0.5+erf((log(a)-mu)/(sqrt(2)*sigma))== 0.01, 0.5+erf((log(b)-mu)/(sqrt(2)*sigma))== 0.99)

x=0:1:150

lgn=lognpdf(x,solx, soly)

plot(x, lgn)

Y = prctile(lgn,0.01)

[the cyclist]

Should be erfc, not erf.

[SYML2nd]

ok, if I do this

a=30
b=60
syms mu sigma 

[solx,soly] = solve( 0.5+erfc((log(a)-mu)/(sqrt(2)*sigma))== 0.01, 0.5+erfc((log(b)-mu)/(sqrt(2)*sigma))== 0.99)

it says Warning: Unable to find explicit solution.

and it gives no solution

[the cyclist]

It should be

0.5*

not

0.5+

[SYML2nd]

It doesn't work. Furthermore looking at the wikipedia section that you posted, it is possible to see that the cfd is 0.5+0.5*erf((log(x)-mu)/(sqrt(2)*sigma))

so I did this code, but when I do the reverse, trying to find the 1st percentile from the built curve the result is not 30.

a=30
b=60
syms mu sigma 

[solx,soly] = solve( 0.5+(0.5*erf((log(a)-mu)/(sqrt(2)*sigma)))== 0.01, 0.5+(0.5*erf((log(b)-mu)/(sqrt(2)*sigma)))== 0.99);
x=0:1:150
lgn=lognpdf(x,solx, soly)
plot(x, lgn)
Y = prctile(lgn,1)

[the cyclist]

I need to think a bit more carefully, but I'm pretty sure the code

x=0:1:150
lgn=lognpdf(x,solx, soly)
plot(x, lgn)
Y = prctile(lgn,1)

is not a valid way to find the 1st percentile of the distribution.

This should give approximately 0.01

logncdf(30,solx,soly)

and

logninv(0.01,solx,soly)

should give 30, if solx and soly are the correct distribution parameters.

[SYML2nd]

Ok. It is correct. Thank you very much