Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
1,035
questions
155
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6answers
86k 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 ...
202
votes
3answers
275k 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 ...
176
votes
8answers
352k views
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?
112
votes
4answers
32k views
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 ...
40
votes
4answers
20k 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 ...
476
votes
14answers
189k views
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 ...
54
votes
5answers
17k 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 ...
41
votes
3answers
6k 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 ...
24
votes
2answers
41k views
How does the inverse transform method work?
How does the inversion method work?
Say I have a random sample $X_1,X_2,...,X_n$ with density $f(x;\theta)={1\over \theta} x^{(1-\theta)\over \theta}$ over
$0<x<1$ and therefore with cdf $F_X(x)=...
28
votes
2answers
1k views
I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?
The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
119
votes
10answers
77k 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?
25
votes
3answers
3k views
Statistical interpretation of Maximum Entropy Distribution
I have used the principle of maximum entropy to justify the use of several distributions in various settings; however, I have yet to be able to formulate a statistical, as opposed to information-...
34
votes
3answers
18k 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 ...
6
votes
2answers
1k views
Simple method of forecasting number of guests given current and historical data
I am trying to predict the number of guests a restaurant might serve in a meal period based on the volume of business that same day from prior years (3-5 years of data), trends for the same day of the ...
39
votes
9answers
9k 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 $\...
152
votes
2answers
205k 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 ...
31
votes
5answers
47k 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 ...
41
votes
4answers
20k 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 ...
32
votes
3answers
23k views
What distribution does my data follow?
Let us say that I have 1000 components and I have been collecting data on how many times these log a failure and each time they logged a failure, I am also keeping track of how long it took my team to ...
44
votes
3answers
15k 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 ...
20
votes
2answers
20k views
What is the definition of a symmetric distribution?
What's the definition of a symmetric distribution? Someone told me that a random variable $X$ came from a symmetric distribution if and only if $X$ and $-X$ has the same distribution. But I think this ...
29
votes
3answers
41k views
Outlier Detection on skewed Distributions
Under a classical definition of an outlier as a data point outide the 1.5* IQR from the upper or lower quartile, there is an assumption of a non-skewed distribution. For skewed distributions (...
134
votes
3answers
184k 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. ...
75
votes
7answers
125k 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 ...
84
votes
5answers
196k 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 ...
32
votes
2answers
35k 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 ...
29
votes
1answer
51k 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 ...
22
votes
2answers
10k views
sum of noncentral Chi-square random variables
I need to find the distribution of the random variable
$$Y=\sum_{i=1}^{n}(X_i)^2$$
where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find ...
27
votes
5answers
45k views
How to sample from a discrete distribution? [duplicate]
Assume I have a distribution governing the possible outcome from a single random variable X.
This is something like [0.1, 0.4, 0.2, 0.3] for X being a value of either 1, 2, 3, 4.
Is it possible to ...
25
votes
3answers
12k views
Student t as mixture of gaussian
Using the student t-distribution with $k > 0$ degrees of freedom, location parameter $l$ and scale parameter $s$ having density
$$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\...
36
votes
2answers
45k 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 ...
39
votes
5answers
22k 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 ...
30
votes
7answers
31k 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%
...
20
votes
2answers
10k views
What is the distribution of the difference of two-t-distributions
... and why ?
Assuming $X_1$,$X_2$ are independent random-variables with mean $\mu_1,\mu_2$ and variance $\sigma^2_1,\sigma^2_2$ respectively. My basic statistics book tells me that the distribution ...
71
votes
4answers
8k 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 ...
23
votes
4answers
30k views
How to test if my distribution is multimodal?
When I plot a histogram of my data, it has two peaks:
Does that mean a potential multi-modal distribution? I ran the dip.test in R (...
10
votes
2answers
9k views
Difference of Gamma random variables
Given two independent random variables $X\sim \mathrm{Gamma}(\alpha_X,\beta_X)$ and $Y\sim \mathrm{Gamma}(\alpha_Y,\beta_Y)$, what is the distribution of the difference, i.e. $D=X-Y$?
If the result ...
13
votes
1answer
1k views
Identity of moment-generating functions
Are there any non-identical distributions which happen to have the same moment-generating function?
7
votes
2answers
9k views
Testing normality
I have a large dataset (500000 data, V1 column include all the data).
x <- read.csv("mydata.csv", header=F)
hist(x)
Which gives:
Looking at the data, I ...
21
votes
5answers
44k views
What is the reason the log transformation is used with right-skewed distributions?
I once heard that
log transformation is the most popular one for right-skewed distributions in linear regression or quantile regression
I would like to know is there any reason underlying this ...
64
votes
3answers
67k 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?
30
votes
2answers
3k 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 ...
30
votes
2answers
40k 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 ...
13
votes
2answers
1k views
Is joint normality a necessary condition for the sum of normal random variables to be normal?
In comments following this answer of mine to a related question, Users ssdecontrol and Glen_b asked
whether joint normality of $X$ and $Y$ is necessary for asserting the
normality of the sum $X+Y$? ...
26
votes
7answers
2k views
Distribution hypothesis testing - what is the point of doing it if you can't “accept” your null hypothesis?
Various hypothesis tests, such as the $\chi^{2}$ GOF test, Kolmogorov-Smirnov, Anderson-Darling, etc., follow this basic format:
$H_0$: The data follow the given distribution.
$H_1$: The data do not ...
23
votes
4answers
5k views
Weakly informative prior distributions for scale parameters
I have been using log normal distributions as prior distributions for scale parameters (for normal distributions, t distributions etc.) when I have a rough idea about what the scale should be, but ...
18
votes
3answers
42k views
How to decide which glm family to use?
I have fish density data that I am trying to compare between several different collection techniques, the data has lots of zeros, and the histogram looks vaugley appropriate for a poisson distribution ...
19
votes
2answers
1k views
Constructing a discrete r.v. having as support all the rationals in $[0,1]$
This is the constructivist sequel of this question.
If we can't have a discrete uniform random variable having as support all the rationals in the interval $[0,1]$, then the next best thing is:
...
5
votes
1answer
6k views
If random variables are drawn from an identical distribution, why doesn't this guarantee they are independent?
Having read a little about exchangeability, I went back to thinking about the iid condition required for the central limit theorem. It struck me that if two random variables are drawn from an ...
12
votes
2answers
39k views
In left skewed data, what is the relationship between mean and median?
I think the median $\leq$ mean.
Is this the case?
