Transform Mean and standard deviation that follow normal distribution into Mi and Sigma that follow Logarithmic normal distribution (Log-Normal)

As the title described, I have a problem with reliability with Mean Time To Repair (MTTR) with Mean = 550 e Standard derivation = 27 hours (Normal distribution). But, The maintenance is better represented as a Lognormal distribution and not Normal Distribution.

How can I convert Mi and Standart Derivation used into Normal Distribution to use it in Lognormal distribution.

Cheers

Answer 1

[Jeff Miller]

finalmu = 550;
finalsigma = 27;
lnsigma = sqrt( log( finalsigma^2 / finalmu^2 + 1) );
lnmu = log( finalmu/exp(0.5*lnsigma^2) );

[yogan sganzerla]

It worked! Cheers! This is my code now:

mu    = 550; % mean for normal distribuction
sigma = 27;  % standard deviation for normal distribuction
lnsigma = sqrt(log( sigma^2 / mu^2 + 1)); % mean for LogNormal 
lnmu = log( mu/exp(0.5*lnsigma^2) );% standard for LogNormal 
Y = normrnd(mu,sigma,1,2*10^3); %randomic values following normal Distribution 
L  = lognrnd(lnmu,lnsigma,1,2*10^3);%randomic values following LogNormal Distribution

What I want is compare a Normal distribution with a Lognormal distribution. To do it I plot using Histogram as shown below.

figure(1);histogram(L,50);
figure(2);histogram(Y,50);

Why the figure(1) and (2) looks like very similar? In my opinion, both look like following a normal distribution.

I am adding the image of Figure 1 and Figure 2

[Jeff Miller]

The amount of skew in the lognormal depends very much on the parameters. With your parameter values, the lognormal has hardly any skew at all, so it looks nearly identical to a normal with the same mean and standard deviation. In essence, this means you would need many, many observations to be able to decide whether the lognormal or normal provided a better fit.

p.s. If my answer helped you, please accept it.

[yogan sganzerla]

Thanks Jeff, I accept your first answer.

I still don't understand the reason that my graphics look like the same. Both have in x_axes the mean equal to 550. What the mean of the second plot isn't the variable "lnmu" that is 6.3087.

I found this link talking about the same problem: https://www.mathworks.com/matlabcentral/answers/80803-generate-a-random-number-from-the-mean-and-standard-deviation-of-a-lognormal-distribution

My other question is: Why the first answer, from Shashank Prasanna, he used this code to convert Mean and std that follow normal distriution into Mean and Std that follow lognormal distribution? This is the code:

    m=31.59299751; 
  s=69.72271594;
  mu = log(m^2/sqrt(s+m^2))
  sigma = sqrt(log(1+s/m^2))
  lognrnd(mu,sigma)
Why it is different that I am using, or they are the same?

Cheers

[Jeff Miller]

> the mean of the second plot isn't the variable "lnmu" that is 6.3087.

This is because the lognormal distribution is a transformation of a normally distributed variable. Specifically, if X is normal, then Y=exp(X) has a lognormal distribution. (That is, the distribution of log(Y) is normal.) In your case, the mean of X is 6.3087. But you are plotting the distribution of Y, and its mean is closer to exp(6.3087)--not exactly the same because exp() is a nonlinear transformation.

If you plotted X=log(Y), then you would see a normal with mean 6.3087 as I guess you are expecting.

I am not sure about the earlier equations. They are not the same, but maybe the earlier s was variance rather than standard deviation?