All Questions
1,289 questions
83
votes
3
answers
105k
views
How is the minimum of a set of IID random variables distributed?
If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
- 941
82
votes
4
answers
60k
views
Linear model with log-transformed response vs. generalized linear model with log link
In this paper titled "CHOOSING AMONG GENERALIZED LINEAR MODELS APPLIED TO MEDICAL DATA" the authors write:
In a generalized linear model, the mean is transformed, by the link
function, instead of ...
- 3,814
73
votes
4
answers
191k
views
How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?
Given the random variable
$$Y = \max(X_1, X_2, \ldots, X_n)$$
where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
- 871
69
votes
9
answers
8k
views
Taleb and the Black Swan
Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual ...
- 43.2k
63
votes
3
answers
25k
views
Which has the heavier tail, lognormal or gamma?
(This is based on a question that just came to me via email; I've added some context from a previous brief conversation with the same person.)
Last year I was told that the gamma distribution is ...
- 290k
52
votes
2
answers
36k
views
Gamma vs. lognormal distributions
I have an experimentally observed distribution that looks very similar to a gamma or lognormal distribution. I've read that the lognormal distribution is the maximum entropy probability distribution ...
- 1,257
32
votes
3
answers
17k
views
Extreme Value Theory - Show: Normal to Gumbel
The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory.
How can we show that?
We have
$$P(\max X_i \leq x) = P(...
- 1,271
32
votes
3
answers
9k
views
Difference of two i.i.d. lognormal random variables
Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$.
The best I can do is to take the Taylor series of both and ...
- 421
29
votes
1
answer
45k
views
Why stock prices are lognormal but stock returns are normal
Except for the fact that returns can be negative while prices must be positive, is there any other reason behind modelling stock prices as a log normal distribution but modelling stock returns as a ...
- 6,635
28
votes
1
answer
49k
views
Expected value and variance of log(a)
I have a random variable $X(a) = \log(a)$ where a is normal distributed $\mathcal N(\mu,\sigma^2)$. What can I say about $E(X)$ and $Var(X)$? An approximation would be helpful too.
- 435
27
votes
5
answers
59k
views
How to specify a lognormal distribution in the glm family argument in R?
Simple question: How to specify a lognormal distribution in the GLM family argument in R?
I could not find how this can be achieved. Why is lognormal (or exponential) not an option in the family ...
- 1,635
27
votes
7
answers
26k
views
How do I calculate a confidence interval for the mean of a log-normal data set?
I've heard/seen in several places that you can transform the data set into something that is normal-distributed by taking the logarithm of each sample, calculate the confidence interval for the ...
- 677
27
votes
6
answers
19k
views
Interpreting the difference between lognormal and power law distribution (network degree distribution)
As part of the network analysis, I plotted a Complementary Cumulative Distribution Function (CCDF) of network degrees. What I found was that, unlike conventional network distributions (e.g. WWW), the ...
- 381
26
votes
6
answers
53k
views
Can mean plus one standard deviation exceed maximum value?
I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33.
My professor asks me how can mean plus one standard deviation exceed the maximum.
I showed her ...
- 261
26
votes
4
answers
46k
views
The sum of independent lognormal random variables appears lognormal?
I'm trying to understand why the sum of two (or more) lognormal random variables approaches a lognormal distribution as you increase the number of observations. I've looked online and not found any ...
- 1,779
26
votes
2
answers
2k
views
Bias of moment estimator of lognormal distribution
I am doing some numerical experiment that consists in sampling a lognormal distribution $X\sim\mathcal{LN}(\mu, \sigma)$, and trying to estimate the moments $\mathbb{E}[X^n]$ by two methods:
Looking ...
- 363
26
votes
1
answer
9k
views
Whether distributions with the same moments are identical
Following are similar to but different from previous posts here and here
Given two distributions which admit moments of all orders, if all the moments of two distributions are the same, then are they ...
- 19.8k
26
votes
1
answer
47k
views
Is a log transformation a valid technique for t-testing non-normal data?
In reviewing a paper, the authors state, "Continuous outcome variables exhibiting a skewed distribution were transformed, using the natural logarithms, before t tests were conducted to satisfy the ...
- 361
25
votes
2
answers
2k
views
Which distribution has its maximum uniformly distributed?
Let's consider $Y_n$ the max of $n$ iid samples $X_i$ of the same distribution:
$Y_n = max(X_1, X_2, ..., X_n)$
Do we know some common distributions for $X$ such that $Y$ is uniformly distributed $U(a,...
- 403
25
votes
5
answers
6k
views
What exactly are moments? How are they derived?
We are typically introduced to method of moments estimators by "equating population moments to their sample counterpart" until we have estimated all of the population's parameters; so that, in the ...
- 1,427
25
votes
2
answers
1k
views
Fitting custom distributions by MLE
My question relates to fitting custom distributions in R but I feel it has enough of a probability element to remain on CV.
I have an interesting set of data which has the following characteristics:
...
- 2,911
24
votes
2
answers
11k
views
Distribution of the maximum of two correlated normal variables
Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly
normal with correlation coefficient $r$.
What is the distribution function of $\max(X_1, X_2)$?
- 2,295
24
votes
5
answers
5k
views
Why use extreme value theory?
I'm coming from Civil Engineering, in which we use Extreme Value Theory, like GEV distribution to predict the value of certain events, like The biggest wind speed, i.e the value that 98.5% of the wind ...
- 1,285
24
votes
6
answers
48k
views
Why doesn't k-means give the global minimum?
I read that the k-means algorithm only converges to a local minimum and not to a global minimum. Why is this? I can logically think of how initialization could affect the final clustering and there is ...
- 589
23
votes
3
answers
3k
views
Distribution of the largest fragment of a broken stick (spacings)
Let a stick of length 1 be broken in $k+1$ fragments uniformly at random. What is the distribution of the length of the longest fragment?
More formally, let $(U_1, \ldots U_k)$ be IID $U(0,1)$, and ...
- 14.9k
21
votes
5
answers
2k
views
Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?
If it is impossible, what is the proof?
- 542
21
votes
2
answers
2k
views
How can we bound the probability that a random variable is maximal?
$\newcommand{\P}{\mathbb{P}}$Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_n$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$....
- 738
19
votes
2
answers
12k
views
What is the variance of the maximum of a sample?
I'm looking for bounds on the variance of the maximum of a set of random variables. In other words, I'm looking for closed-form formulas for $B$, such that
$$
\mbox{Var}(\max_i X_i) \leq B \enspace,
$$...
- 273
18
votes
1
answer
22k
views
Correlation of log-normal random variables
Given $X_1$ and $X_2$ normal random variables with correlation coefficient $\rho$, how do I find the correlation between following lognormal random variables $Y_1$ and $Y_2$?
$Y_1 = a_1 \exp(\mu_1 T +...
- 2,799
17
votes
1
answer
3k
views
Predicting y from log y as the dependent variable
In the book Introductory Econometrics by Wooldridge the chapter, which deals with predicting values of $\hat{y}$ (chapter 6.4 in the 5th edition) states the following:
If the estimated model is: ...
- 451
15
votes
2
answers
21k
views
What is the distribution for the maximum (minimum) of two independent normal random variables?
Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...
- 355
15
votes
1
answer
10k
views
Finding local extrema of a density function using splines
I am trying to find the local maxima for a probability density function (found using R's density method). I cannot do a simple "look around neighbors" method (where ...
- 373
15
votes
1
answer
2k
views
Why is the arithmetic mean smaller than the distribution mean in a log-normal distribution?
So, I have a random process generating log-normally distributed random variables $X$. Here is the corresponding probability density function:
I wanted to estimate the distribution of a few moments of ...
- 830
15
votes
3
answers
21k
views
When is it OK to write "we assumed a normal distribution" of an empirical measurement?
It is ingrained in the teaching of applied disciplines, such as medicine, that measurements of bio-medical quantities in the population follow a normal "bell curve." A Google search of the the string "...
- 26.9k
14
votes
1
answer
8k
views
Why is ln[E(x)] > E[ln(x)]?
We're dealing with the lognormal distribution in a finance course and my textbook just states that this is true, which I find sort of frustrating as my maths background isn't very strong but I want ...
- 173
14
votes
1
answer
1k
views
Any example of (roughly) independent variables that are dependent at extreme values?
I am looking for an example of 2 random variables $X$, $Y$ such that
$$\newcommand{\cor}{{\rm cor}}|\cor(X,Y)| \approx 0 $$
but when consider the tail part of the distributions, they are highly ...
- 143
14
votes
4
answers
1k
views
Unbiased estimator for the smaller of two random variables
Suppose $X \sim \mathcal{N}(\mu_x, \sigma^2_x)$ and $Y \sim \mathcal{N}(\mu_y, \sigma^2_y)$
I am interested in $z = \min(\mu_x, \mu_y)$. Is there an unbiased estimator for $z$?
The simple estimator ...
- 141
14
votes
3
answers
13k
views
Priors for log-normal models
I am trying to determine what the most appropriate non-informative priors are for the two parameters of a log-normal distribution (for an accelerated failure time model). I had been working with a ...
- 729
14
votes
2
answers
1k
views
How to determine the distribution of a parameter fit by nonlinear regression
The example above shows enzyme kinetics -- enzyme velocity as a function of substrate concentration. The well-established Michaelis-Menten equation is:
$Y=V_{max} \cdot \dfrac{X}{K_m + X}$
$X$ are ...
- 21.2k
14
votes
1
answer
2k
views
Does a median-unbiased estimator minimize mean absolute deviance?
This is a follow-up but also a different question of my previous one.
I read on Wikipedia that "A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as ...
- 393
13
votes
3
answers
2k
views
Need algorithm to compute relative likelihood that data are sample from normal vs lognormal distribution
Let's say you have a set of values, and you want to know if it is more likely that they were sampled from a Gaussian (normal) distribution or sampled from a lognormal distribution?
Of course, ...
- 21.2k
13
votes
1
answer
15k
views
Expectation, Variance and Correlation of a bivariate Lognormal distribution
If $Y \sim N(\mu,\sigma^2)$ is normally distributed, then $X=\mathrm{e}^Y$ is lognormally distributed. To get the log-$\mu$ and log-$\sigma$ of this lognormal distribution you calculate
$$\sigma^2 = \...
- 1,781
13
votes
1
answer
15k
views
Multivariate log-normal probabiltiy density function (PDF)
The Multivariate Gaussian pdf is given by
$$(2\pi)^{-\frac{K}{2}} \det(\Sigma)^{-\frac{1}{2}} \exp({-\frac{1}{2}}(X-\mu)' \Sigma^{-1} (X-\mu)) $$
The wikipedia for multivariate Gaussians is here
...
- 1,235
13
votes
2
answers
9k
views
Markov chain Monte Carlo (MCMC) for Maximum Likelihood Estimation (MLE)
I am reading a 1991 conference paper by Geyer which is linked below. In it he seems to elude to a method that can use MCMC for MLE parameter estimation
This excites me since, I have coded BFGS ...
13
votes
3
answers
1k
views
Does there exist someone faster than Usain Bolt today?
EDIT: I am more interested in the technical issues and methodology of determining the likelihood of a "true" maximum in a given population given a sample statistic. There are problems with estimating ...
- 283
12
votes
3
answers
35k
views
Exponential of a standard normal random variable
We know that $Z\sim N(0, 1)$. How do I prove that $e^Z$ has a mean of $e^{0.5}$? I have tried integrating $e^z$ times the pdf of $Z$ but I don't know why it isn't working out.
Also what is the pdf ...
- 121
12
votes
3
answers
32k
views
Calculating distribution from min, mean, and max
Suppose I have the minimum, mean, and maximum of some data set, say, 10, 20, and 25. Is there a way to:
create a distribution from these data, and
know what percentage of the population likely lies ...
- 123
12
votes
3
answers
1k
views
Classes of distributions closed under maximum
Let $Q_p$ be a class of probability distributions on non-negative reals parametrized by $p$, so that
$$
Q_p([0,\infty)) = 1.
$$
I wonder which known classes of distributions are closed under ...
- 473
12
votes
1
answer
1k
views
Card game: If I draw four cards randomly and you draw six, what is the probability that my highest card is higher than your highest?
As stated in the title, say if I draw randomly 4 cards and you draw 6 from the same deck, what is the probability that my highest card beats your highest card?
How will this change if we draw from ...
- 611
12
votes
1
answer
11k
views
Is it possible to analytically integrate $x$ multiplied by the lognormal probability density function?
Firstly, by analytically integrate, I mean, is there an integration rule to solve this as opposed to numerical analyses (such as trapezoidal, Gauss-Legendre or Simpson's rules)?
I have a function $\...
- 123