Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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530 votes
15 answers
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What is the intuition behind beta distribution?

Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial ...
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224 votes
4 answers
326k views

When (and why) should you take the log of a distribution (of numbers)?

Say I have some historical data e.g., past stock prices, airline ticket price fluctuations, past financial data of the company... Now someone (or some formula) comes along and says "let's take/use ...
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205 votes
8 answers
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In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values?

Am I looking for a better behaved distribution for the independent variable in question, or to reduce the effect of outliers, or something else?
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186 votes
2 answers
255k views

How to determine which distribution fits my data best?

I have a dataset and would like to figure out which distribution fits my data best. I used the fitdistr() function to estimate the necessary parameters to ...
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174 votes
6 answers
108k views

Can a probability distribution value exceeding 1 be OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
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148 votes
3 answers
217k views

Help me understand Bayesian prior and posterior distributions

In a group of students, there are 2 out of 18 that are left-handed. Find the posterior distribution of left-handed students in the population assuming uninformative prior. Summarize the results. ...
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130 votes
10 answers
96k views

Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
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117 votes
4 answers
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Assessing approximate distribution of data based on a histogram

Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right). Depending on how I group or bin the data, I can get wildly different histograms. One set of ...
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98 votes
9 answers
69k views

Understanding "variance" intuitively

What is the cleanest, easiest way to explain someone the concept of variance? What does it intuitively mean? If one is to explain this to their child how would one go about it? It's a concept that I ...
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90 votes
5 answers
233k views

Relationship between poisson and exponential distribution

The waiting times for poisson distribution is an exponential distribution with parameter lambda. But I don't understand it. Poisson models the number of arrivals per unit of time for example. How is ...
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  • 2,349
90 votes
7 answers
155k views

Calculating the parameters of a Beta distribution using the mean and variance

How can I calculate the $\alpha$ and $\beta$ parameters for a Beta distribution if I know the mean and variance that I want the distribution to have? Examples of an R command to do this would be most ...
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80 votes
4 answers
12k views

What's so 'moment' about 'moments' of a probability distribution?

I KNOW what moments are and how to calculate them and how to use the moment generating function for getting higher order moments. Yes, I know the math. Now that I need to get my statistics knowledge ...
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72 votes
5 answers
24k views

Intuition on the Kullback–Leibler (KL) Divergence

I have learned about the intuition behind the KL Divergence as how much a model distribution function differs from the theoretical/true distribution of the data. The source I am reading goes on to say ...
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72 votes
3 answers
85k 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?
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65 votes
5 answers
25k views

What is the advantages of Wasserstein metric compared to Kullback-Leibler divergence?

What is the practical difference between Wasserstein metric and Kullback-Leibler divergence? Wasserstein metric is also referred to as Earth mover's distance. From Wikipedia: Wasserstein (or ...
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58 votes
6 answers
47k views

How can a distribution have infinite mean and variance?

It would be appreciated if the following examples could be given: A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with ...
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57 votes
13 answers
170k views

Mean absolute deviation vs. standard deviation

In the text book "New Comprehensive Mathematics for O Level" by Greer (1983), I see averaged deviation calculated like this: Sum up absolute differences between single values and the mean. Then ...
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55 votes
4 answers
43k views

How to identify a bimodal distribution?

I understand that once we plot the values as a chart, we can identify a bimodal distribution by observing the twin-peaks, but how does one find it programmatically? (I am looking for an algorithm.)
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52 votes
6 answers
250k views

How to perform a test using R to see if data follows normal distribution

I have a data set with following structure: a word | number of occurrence of a word in a document | a document id How can I perform a test for normal ...
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52 votes
3 answers
19k 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 ...
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52 votes
4 answers
65k views

If the t-test and the ANOVA for two groups are equivalent, why aren't their assumptions equivalent?

I'm sure I've got this completely wrapped round my head, but I just can't figure it out. The t-test compares two normal distributions using the Z distribution. That's why there's an assumption of ...
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  • 5,505
52 votes
4 answers
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Kullback–Leibler vs Kolmogorov-Smirnov distance

I can see that there are a lot of formal differences between Kullback–Leibler vs Kolmogorov-Smirnov distance measures. However, both are used to measure the distance between distributions. Is there a ...
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50 votes
4 answers
26k views

Approximate order statistics for normal random variables

Are there well known formulas for the order statistics of certain random distributions? Particularly the first and last order statistics of a normal random variable, but a more general answer would ...
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50 votes
5 answers
34k views

Probability distribution for different probabilities

If I wanted to get the probability of 9 successes in 16 trials with each trial having a probability of 0.6 I could use a binomial distribution. What could I use if each of the 16 trials has a ...
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50 votes
6 answers
14k views

Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
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49 votes
5 answers
29k views

Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
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49 votes
3 answers
10k views

How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things ...
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48 votes
8 answers
64k views

How can I test if given samples are taken from a Poisson distribution?

I know of normality tests, but how do I test for "Poisson-ness"? I have sample of ~1000 non-negative integers, which I suspect are taken from a Poisson distribution, and I would like to test that.
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  • 1,191
46 votes
5 answers
71k views

How does linear regression use the normal distribution?

In linear regression, each predicted value is assumed to have been picked from a normal distribution of possible values. See below. But why is each predicted value assumed to have come from a normal ...
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46 votes
3 answers
19k views

Empirical relationship between mean, median and mode

For a unimodal distribution that is moderately skewed, we have the following empirical relationship between the mean, median and mode: $$ \text{(Mean - Mode)}\sim 3\,\text{(Mean - Median)} $$ How ...
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  • 1,367
43 votes
5 answers
5k views

Fake uniform random numbers: More evenly distributed than true uniform data

I'm looking for a way to generate random numbers that appear to be uniform distributed -- and every test will show them to be uniform -- except that they are more evenly distributed than true uniform ...
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43 votes
3 answers
75k views

Help me understand the quantile (inverse CDF) function

I am reading about the quantile function, but it is not clear to me. Could you provide a more intuitive explanation than the one provided below? Since the cdf $F$ is a monotonically increasing ...
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43 votes
2 answers
57k views

Test for bimodal distribution

I wonder if there is any statistical test to "test" the significance of a bimodal distribution. I mean, How much my data meets the bimodal distribution or not? If so, is there any test in the R ...
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  • 613
43 votes
2 answers
31k views

Understanding the parameters inside the Negative Binomial Distribution

I was trying to fit my data into various models and figured out that the fitdistr function from library MASS of ...
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42 votes
2 answers
48k views

What exactly is the alpha in the Dirichlet distribution?

I'm fairly new to Bayesian statistics and I came across a corrected correlation measure, SparCC, that uses the Dirichlet process in the backend of it's algorithm. I have been trying to go through the ...
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42 votes
6 answers
14k views

Intuitive explanation of convergence in distribution and convergence in probability

What is the intuitive difference between a random variable converging in probability versus a random variable converging in distribution? I've read numerous definitions and mathematical equations, ...
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  • 1,163
42 votes
1 answer
75k views

Why is the sampling distribution of variance a chi-squared distribution?

The statement The sampling distribution of the sample variance is a chi-squared distribution with degree of freedom equals to $n-1$, where $n$ is the sample size (given that the random variable of ...
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  • 4,642
41 votes
9 answers
12k views

How can I efficiently model the sum of Bernoulli random variables?

I am modeling a random variable ($Y$) which is the sum of some ~15-40k independent Bernoulli random variables ($X_i$), each with a different success probability ($p_i$). Formally, $Y=\sum X_i$ where $\...
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  • 1,191
40 votes
2 answers
4k views

Why should we use t errors instead of normal errors?

In this blog post by Andrew Gelman, there is the following passage: The Bayesian models of 50 years ago seem hopelessly simple (except, of course, for simple problems), and I expect the Bayesian ...
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  • 1,035
40 votes
3 answers
33k views

Why is a likelihood-ratio test distributed chi-squared?

Why is the test statistic of a likelihood ratio test distributed chi-squared? $2(\ln \text{ L}_{\rm alt\ model} - \ln \text{ L}_{\rm null\ model} ) \sim \chi^{2}_{df_{\rm alt}-df_{\rm null}}$
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39 votes
10 answers
9k views

Why are survival times assumed to be exponentially distributed?

I am learning survival analysis from this post on UCLA IDRE and got tripped up at section 1.2.1. The tutorial says: ... if the survival times were known to be exponentially distributed, then the ...
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39 votes
5 answers
17k views

Is there an explanation for why there are so many natural phenomena that follow normal distribution?

I think this is a fascinating topic and I do not fully understand it. What law of physics makes so that so many natural phenomena have normal distribution? It would seem more intuitive that they would ...
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  • 559
39 votes
3 answers
26k views

Why is RSS distributed chi square times n-p?

I would like to understand why, under the OLS model, the RSS (residual sum of squares) is distributed $$\chi^2\cdot (n-p)$$ ($p$ being the number of parameters in the model, $n$ the number of ...
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39 votes
2 answers
135k views

What is the difference between the Shapiro–Wilk test of normality and the Kolmogorov–Smirnov test of normality?

What is the difference between the Shapiro–Wilk test of normality and the Kolmogorov–Smirnov test of normality? When will results from these two methods differ?
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38 votes
4 answers
32k views

Measures of similarity or distance between two covariance matrices

Are there any measures of similarity or distance between two symmetric covariance matrices (both having the same dimensions)? I am thinking here of analogues to KL divergence of two probability ...
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37 votes
4 answers
4k views

Intuitive explanation of Kolmogorov Smirnov Test

What is the cleanest, easiest way to explain someone the concept of Kolmogorov Smirnov Test? What does it intuitively mean? It's a concept that I have difficulty in articulating - especially when ...
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  • 2,391
37 votes
4 answers
29k views

How does one measure the non-uniformity of a distribution?

I'm trying to come up with a metric for measuring non-uniformity of a distribution for an experiment I'm running. I have a random variable that should be uniformly distributed in most cases, and I'd ...
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  • 473
36 votes
3 answers
5k views

I know the 95% confidence interval for ln(x), do I also know the 95% confidence interval of x?

Suppose the 95% confidence interval for $\ln(x)$ is $[l,u]$. Is it true that the 95% CI for $x$ is simply $[e^l, e^u]$? I have the intuition the answer is yes, because $\ln$ is a continuous function. ...
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  • 485
36 votes
6 answers
3k views

How can I analytically prove that randomly dividing an amount results in an exponential distribution (of e.g. income and wealth)?

In this current article in SCIENCE the following is being proposed: Suppose you randomly divide 500 million in income among 10,000 people. There's only one way to give everyone an equal, 50,000 ...
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  • 5,936
35 votes
7 answers
35k views

How to generate numbers based on an arbitrary discrete distribution?

How do I generate numbers based on an arbitrary discrete distribution? For example, I have a set of numbers that I want to generate. Say they are labelled from 1-3 as follows. 1: 4%, 2: 50%, 3: 46% ...
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