# Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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### 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 ...
256k 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 ...
331k 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?
31k 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 ...
18k 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 ...
180k 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 ...
15k 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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### 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 ...
17k 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 ...
42k 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 ...
22k 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 ...
14k 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 ...
19k 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 ...
176k 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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### 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 ...
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 ...
189k 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 ...
37k 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 (...
117k 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 ...
46k 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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### 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 ...
42k 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 ...
43k 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 ...
11k 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}\...
9k 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 ...
7k 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 ...
31k 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 ...
21k 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 ...
8k 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 ...
29k 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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### Identity of moment-generating functions

Are there any non-identical distributions which happen to have the same moment-generating function?
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 ...
34k 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 ...
62k views

### How is the minimum of a set of 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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### 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 ...
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 ...
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### 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 ...
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 ...
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### 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 (...
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### 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 $X+Y$? ...
5k 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 ...
38k views

### In left skewed data, what is the relationship between mean and median?

I think the median $\leq$ mean. Is this the case?
### $X_i, X_j$ independent when $i≠j$, but $X_1, X_2, X_3$ dependent?
I've seen the statement: It's possible that random variables $X_i, X_j$ are independent for $i≠j$, but $X_1, X_2, X_3$ are dependent. I haven't been able to find examples of this though. Any ...