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 ...
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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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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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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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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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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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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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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 ...
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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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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)=...
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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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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-...
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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 ...
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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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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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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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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 ...
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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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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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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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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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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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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 ...
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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 ...
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What are the differences between stochastic and fixed regressors in linear regression model?
If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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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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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}\...
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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 ...
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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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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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"Absolutely continuous random variable" vs. "Continuous random variable"?
In the book Limit Theorems of Probability Theory by Valentin V. Petrov, I saw a distinction between the definitions of a distribution being "continuous" and "absolutely continuous",...
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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 ...
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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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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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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 (...
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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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A normal divided by the $\sqrt{\chi^2(s)/s}$ gives you a t-distribution -- proof
let $Z \sim N(0,1)$ and $W \sim \chi^2(s)$.
If $Z$ and $W$ are independently distributed then the variable $Y = \frac{Z}{\sqrt{W/s}}$ follows a $t$ distribution with degrees of freedom $s$.
I am ...
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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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Identity of moment-generating functions
Are there any non-identical distributions which happen to have the same moment-generating function?
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$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 ...
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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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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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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:
...
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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$? ...
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How does entropy depend on location and scale?
The entropy of a continuous distribution with density function $f$ is defined to be the negative of the expectation of $\log(f),$ and therefore equals
$$H_f = -\int_{-\infty}^{\infty} \log(f(x)) f(x)\...
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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 ...
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Probability theory books for self-study
Are there any good books that explain important concepts of probability theory like probability distribution functions and cumulative distribution functions?
Please, avoid referring books like "...
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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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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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What are good data visualization techniques to compare distributions?
I am writing my PhD thesis and I've realized that I rely excessively in box plots in order to compare distributions. Which other alternatives do you like for achieving this task?
I'd also like to ask ...