Talk:Log-normal distribution

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1 Old talk
2 Could I ask a question?
3 A mistake?
4 A Typo
5 Erf and normal cdf
6 Technical
7 Skewness formual incorrect?
8 X log normal, not normal.
9 Partial expectation
10 Generalize distribution of product of lognormal variables

[edit] Old talk

Hello. I have changed the intro from "log-normal distributions" to "log-normal distribution". I do understand the notion that, for each pair of values (mu, sigma), it is a different distribution. However, common parlance is to call all the members of a parametric family by a collective name -- normal distribution, beta distribution, exponential distribution, .... In each case these terms denote a family of distributions. This causes no misunderstandings, and I see no advantage in abandoning that convention. Happy editing, Wile E. Heresiarch 03:42, 8 Apr 2004 (UTC)

In the formula for the maximum likelihood estimate of the logsd, shouldn't it be over n-1, not n?

Unless you see an error in the math, I think its ok. The n-1 term usually comes in when doing unbiased estimators, not maximum likelihood estimators.

You're right; I was confused.

QUESTION: Shouldn't there be a square root at the ML estimation of the standard deviation? User:flonks

Right - I fixed it, thanks. PAR 09:15, 27 September 2005 (UTC)

[edit] Could I ask a question?

If Y=a^2; a is a log normal distribution ; then What kind of distribution is Y?

a is a lognormal distribution

so log(a) is a normal distribution

log(a^2) = 2 log(a) is also a normal distribution

a^2 is a lognormal distribution --Buglee 00:47, 9 May 2006 (UTC)

One should say rather that a has---not is---a lognormal distribution. The object called a is a random variable, not a probability distribution. Michael Hardy 01:25, 9 May 2006 (UTC)

Maria 13 Feb 207: I've never written anything in wikipedia, so I apologise if I am doing the wrong thing. I wanted to note that the following may not be clear to the reader: in the formulas, E(X)^2 represents the square of the mean, rather than the second moment. I would suggest one of the following solutions: 1) skip the parentheses around X and represent the mean by EX. Then it is clear that (EX)^2 will be its square. However, one might wonder about EX^2 (which should represent the second moment...) 2) skip the E operator and put a letter there, i.e. let m be the mean and s the standard deviation. Then there will be no confusion. 3) add a line at some point in the text giving the notation: i.e. that by E(X)^2 you mean the square of the first moment, while the second moment is denoted by E(X^2) (I presume). I had to invert the formula myself in order to figure out what it is supposed to mean.

I've just attended to this. Michael Hardy 00:52, 14 February 2007 (UTC)

[edit] A mistake?

I think there is a mistake here : the density function should include a term in sigma squared divided by two, and the mean of the log normal variable becomes mu - sigma ^2/2 Basically what happened is that, I think, the author forgot the Ito term.

I believe the article is correct. See for example http://mathworld.wolfram.com/LogNormalDistribution.html for an alternate source of the density function and the mean. They are the same as shown here, but with a different notation. (M in place of mu and S in place of sigma). Encyclops 00:23, 4 February 2006 (UTC)

Either the graph of the density function is wrong, or the expected value formula is wrong. As you can see from the graph, as sigma decreases, the expected value moves towards 1 from below. This is consistent with the mean being exp(mu - sigma^2/2), which is what I recall it as. 69.107.6.4 19:29, 5 April 2007 (UTC)

Here's you're mistake. You cannot see the expected value from the graph at all. It is highly influenced by the fat upper tail, which the graph does not make apparent. See also my comments below. Michael Hardy 20:19, 5 April 2007 (UTC)

I've just computed the integral and I get

$e^{\mu + \sigma^2/2}.\,$

So with μ = 0, as σ decreases to 0, the expected value decreases to 1. Thus it would appear that the graph is wrong. Michael Hardy 19:57, 5 April 2007 (UTC)

...and now I've done some graphs by computer, and they agree with what the illustration shows. More later.... Michael Hardy 20:06, 5 April 2007 (UTC)

OK, there's no error. As the mode decreases, the mean increases, because the upper tail gets fatter! So the graphs and the mean and the mode are correct. Michael Hardy 20:15, 5 April 2007 (UTC)

You're right. My mistake. The mean is highly influenced by the upper tail, so the means are actually decreasing to 1 as sigma decreases. It just looks like the means approach from below because the modes do. 71.198.244.61 23:50, 7 April 2007 (UTC)

[edit] A Typo

There is a typo in the PDF formula, a missing '['

[edit] Erf and normal cdf

There are formulas that use Erf and formulas that use the cdf of the normal distribution, IMHO this is confusing, because those functions are related but not identical. Albmont 15:02, 23 August 2006 (UTC)

[edit] Technical

Please remember that Wikipedia articles need to be accessible to people like high school studends, or younger, or without any background in math. I consider myself rather knowledgable in math (had it at college level, and still do) but (taking into account English is not my native language) I found the lead to this article pretty difficult. Please make it more accessible.--_{Piotr Konieczny aka Prokonsul Piotrus | talk} 22:48, 31 August 2006 (UTC)

To expect all Wikipedia math articles to be accessible to high-school students is unreasonable. Some can be accessible only to mathematicians; perhaps more can be accessible to a broad audience of professionals who use mathematics; others to anyone who's had a couple of years of calculus and no more; others to a broader audience still. Anyone who knows what the normal distribution is, what a random variable is, and what logarithms are, will readily understand the first sentence in this article. Can you be specific about what it is you found difficult about it? Michael Hardy 23:28, 31 August 2006 (UTC)

I removed the "too technical" tag. Feel free to reinsert it, but please leave some more details about what specifically you find difficult to understand. Thanks, Lunch 22:18, 22 October 2006 (UTC)

[edit] Skewness formual incorrect?

The formula for the skewness appears to be incorrect: the leading exponent term you have is not present in the definitions given by Mathworld and NIST, see http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htm and http://mathworld.wolfram.com/LogNormalDistribution.html.

Many thanks.

[edit] X log normal, not normal.

I think the definition of X as normal and Y as lognormal in the beginning of the page should be changed. The rest of the page treats X as the log normal variable. —The preceding unsigned comment was added by 213.115.25.62 (talk) 17:40, 2 February 2007 (UTC).

[edit] Partial expectation

I think that there was a mistake in the formula for the partial expectation: the last term should not be there. Here is a proof: http://faculty.london.edu/ruppal/zenSlides/zCH08%20Black-Scholes.slide.doc See Corollary 2 in Appendix A 2.

I did put my earlier correction back in. Of course, I may be wrong (but, right now, I don't see why). If you change this again, please let me know why I was wrong. Thank you.

Alex —The preceding unsigned comment was added by 72.255.36.161 (talk) 19:39, 27 February 2007 (UTC).

Thanks. I see the problem. You have the correct expression for

$g(k)=\int_k^\infty x f(x)\, dx$

while what I had there before would be correct if we were trying to find

$g_2(k)=\int_k^\infty (x-k) f(x)\, dx$

which is (essentially) the B-S formula but is not the partial mean (or partial expectation) by my (or your) definition. (Actually I did find a few sources where the partial expectation is defined as $g 2$ but this usage seems to be rare. For ex. [1]). The term that you dropped occurs in $g 2 (k)$ but not $g (k)$ , the correct form of the partial mean. So I will leave the formula as it is now. Encyclops 00:47, 28 February 2007 (UTC)

[edit] Generalize distribution of product of lognormal variables

About the distribution of a product of independent log-normal variables:

Wouldn't it be possible to generalize it to variables with different average ( mu NOT the same for every variable)?

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