Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
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Is it useful to define the hazard function of distributions with support in the real line?
The hazard function is commonly used in models using distributions with positive support (gamma, weibull, lognormal, etcetera). However, I have not seen this concept (hazard) being used in the context ...
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How can I fit a distribution to a dataset while forcing it through an exact point in r?
This code was kindly recommended to me in my original question. It returned the same parameter estimates as the software called CRAFT by Aon Benfield. I have also managed to replicate it for the ...
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Why does dbeta not sum to 1?
Both dpois and dnorm in the code below sum to 1 (or thereabouts). This appears to confirm my understanding of the ...
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Is there any advanced version of Total Time on Test (TTT) Transform?
One parameter survival distribution with increasing hazard rate is (for example) Rayleigh and Lindley, where Rayleigh's hazard rate increases linearly and Lindley's hazard rate increases with a ...
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Run lengths in a binary string
Among the binary strings of length $n$, what is the distribution of the lengths of the homogeneous runs ?
E.g., for $n=4$ the possible strings and run lengths are
$$0000: 4;0001: 1,3;0010: 1^2,2;0011:...
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Interpreting box plots with categorical variables
How do you explain a box plot with categorical variables on the x-axis? For example, I have these two box plots, how do you interpret relative comparison of each category within the box plot?
Sample ...
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If $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$ [closed]
Show that if $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$. Can the independence condition be dropped?
I could solve the first part using ...
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$\mathbb E[|X_n|^r]<\infty$ and $\mathbb E[|X_n|^r]\to \mathbb E[|X|^r]$ as $n\to \infty$
Let $\{X_n\}\xrightarrow{d}X$ and for some $p>0$, we have
$$\sup_{n\ge 1} \mathbb E[|X_n|^p]<\infty$$
Show that for any $r\in (0,p)$, we have
a. $\mathbb E[|X|^r]<\infty$
b. $\mathbb E[|X_n|^...
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Checking the Lindeberg Condition
Let $\{X_n\}$ be a sequence of independent random variables such that $\mathbb P(X_n=\pm 1)=\frac 14$, $\mathbb P(X_n=\pm n)=\frac 1{4n^2}$ and $\mathbb P(X_n=0)=\frac 12 - \frac 1{2n^2}$ for all $n\...
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Distribution of Squared Euclidean Norm of Gaussian Vector
If $\boldsymbol{X} \sim \mathcal{N}_N(\boldsymbol{\mu},\boldsymbol{\Sigma})$ is an $N$-dimensional gaussian vector, where $\boldsymbol{\mu} \in \mathbb{R}^N$ and $\boldsymbol{\Sigma} \in \mathbb{R}^{...
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How to combine independent probability distributions?
I have 3 probability distributions derived from three random variables, where the distributions and the variables are independent. How do I combine all 3 distributions, either via direct aggregation ...
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Win probability of runners in a race
This is a problem discussed here before.
We know the mean times of the runners (μ(i)) and the standard deviations (σ(i))) and we want to find the probabilities p(1), p(2) ... p(N).
Here is a solution:
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Counts in ordered categories as response
Trying to figure out the most appropriate family for a data type I'm not used to.
For each measurement, I have a bunch of attempts, with the outcomes falling into bins: failure, small, medium or large....
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How to transform one distribution into another using a linear transform of Mean and linear transform of Std?
I'm trying to account for a time correction between two streams of data, let's say Stream A and Stream B. Stream A and Stream B each have different message arrival/latency characteristics. Each ...
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sum of noncentral Chi random variables
if $X_1,...,X_n$ are independent random variables with noncentral chi distributions (same $df$ but different $\lambda$),
What is the distribution of $\sum_{i=1}^{n}{X_i}$
Just wondering if it can be ...
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Difference between charts of rcauchy(10000) and geom_function(fun = dcauchy)
In R, suppose you generate $10^7$ random variates of the standard Cauchy distribution. Then you plot a histogram and density of this simulated data with the actual standard Cauchy density on top of it....
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Why does one interested in modelling the distribution of median of random samples?
I was reading this article which introduce the families of distribution for the median of an random sample. However, the writer did not state the reason why one is interested in modelling the ...
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How to deal with non-normally distributed data?
I have a proportion data (0-100%, or it can be count data 0-20) that is not normally distributed. The hist graph in R shows:
I have tried many ways to transfer data (log, coxbox, ranktransfer), but ...
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Estimating distribution of minute averages from week averages
I would like to estimate the distribution of 1-minute averages for a process. I have sixty-five 1-week averages to work with. I also have the peak 2-minute averages for those sixty-five weeks.
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Analytical form of a Histogram of $1-\cos(X)$, $X\sim U(0, 2\pi)$
If I generate a uniform distribution of $X$ ranging from $0$ to $2\pi$ (so $X\sim U(0, 2\pi)$), then the probability distribution of $1-\cos(X)$ appears to be this function:
Is this an analytical ...
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Clarifications on parameters in models and distribution
I'm looking for help clarifying the concept of parameters in statistics. Mainly, I'm having trouble understanding the commonality or difference between distribution parameter and model parameter.
When ...
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how to weight means based on data set size
Let's say I have a list of lists of integers, all of a large variance in length, and I want to get the weighted mean of each record. How would I go about doing that?
For example:
...
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Null Hypothesis in Paired t-test
Suppose that I was tasked to find if the mean difference between the current and starting salary of employees is greater than 15,000. How will my null and alternative hypotheses be?
Can it be:
Ho: ...
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Find pdf of $W = X/(X+Y)$ - Result check
I come up with this problem and I am sure if my solution is corrected.
$X, Y \sim \text{Exp}(\lambda)$. Find pdf of $W = X/(X+Y)$.
Please see my try bellow:
At Eq(17) I don't know if my result is ...
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Transforming one distribution into another with different support
Consider the following two random variables:
A random vector $S$ with law $h$ and support $\mathcal{S}$ and,
a random vector $X$ with law $c$ and support $\mathcal{X}$.
Assume
$\text{Dim}(\mathcal{...
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Why is the quotient distribution of two probability distributions angular?
Background
I am working with posterior probability distributions for parameters obtained from a Bayesian binomial generalised linear model with a logit link function. The parameters returned by the ...
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Understanding of bivariate Gaussian distributions in connection with complex random variable
Say that we want to model a complex-valued signal using the RV $S$, where $S$ can be expressed by it's real and imaginary part, i.e. $S = X + iY$, where $X$ and $Y$ are real-valued random variables.
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Statistics Interview Question
Imagine you are solving difficult Math problems and you expect to solve one every 1/2 hour. Compute the probability that you will have to wait between 2 to 4 hours before you solve four of them.
I ...
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How do I adjust the arguments of the Hartigans Dip Test in R to fit my specific data?
I want to test my data (a distribution of saccade-amplitudes, N = 25092) for unimodality with the Hartigans Dip Test, more specific the dip.test() function in the diptest-package in R. A paper I read ...
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Sampling from a preference table [closed]
Suppose we have a table of a group of friend's preferences for food. Each person will receive a unique meal. What is the approach to sample from this preference table so that each time this group ...
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Calculating the distribution of $X-Y$
One can find the distribution of $X+Y$ where $X$ and $Y$ are independent random variables using this formula
$$f_{X+Y}(a)=\int_{-\infty}^\infty f_X(a-y) f_Y(y) dy$$
I'm wondering how to adapt this ...
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I am unsure whether this is a binomial or normal distribution question and how to go about solving it?
The question "Develop a hypothesis, and constructed appropriate null and alternative/experimental hypothesis.
a) Hyacinth bulbs are sold to a retailer in packs of 100 which claim to have equal ...
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Continuous or Discrete Sample Data?
I am currently trying to estimate stock levels in a monte carlo simulation, i have certain integer sample data for product A,B,C with each having a sample size of 100. When trying to do a fit to a ...
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Is every probability distribution also the distribution of the maximum of several i.i.d. random variables?
I found the following result used in this paper, but it was just claimed without proof and it seems extremely strong to me, so I would like a proof, or at least a reference, of a proof.
Let $D$ be ...
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Hypergeometric CDF in R
I'm working on a reliability problem, and I've run into a snag. I have a population of 300 widgets. From that population, I took a random sample of 10 and 1 turned out to be bad. What I'd like to know ...
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Sampling from share transition of population distribution
Dears,
I have this data below, which represents a distribution of population across 8 groups in time 1 and consequent time 2. (time1 > time2). I would like to sample using these 2 inputs, it this ...
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Are there any differences in R if I use a "pnorm" or create a distribution with rnorm for Kolmogorov-Smirnov?
For testing distributions existing in base R, say normal or beta distributions, is there any difference if I use a one-sampled or two-sampled Kolmogorov-Smirnov test with same parameters?
For example:
...
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Does higher variance usually mean lower probability density?
Does higher variance usually mean lower probability density? Despite the type of distribution. Thank you.
Update：
Sorry for confusion. Please allow me to clarify. If I sample the same number of data ...
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Need help finding a corresponding distribution
I made a Monte Carlo based hypothesis test, which starts comparing N values in M simulations to a confidence interval, giving me a binary N*M matrix. Then I calculate the percent of values equal to 1 ...
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Upper bound on Kolmogorov-Smirnov distance after some transformation $h$
Problem setup: Suppose $X_1, \ldots, X_n$ is an i.i.d. sample from $F_X$ (CDF), and $Y_1, \ldots, Y_n$ is another i.i.d. sample from $F_Y$ (also CDF). In addition, $h(z_1, \ldots, z_n)$ is a real-...
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Why is Rectangular density kernel not cut off at tails?
When we create kernel densities we could use different kernels. Here I create an example with Gaussian, Rectangular and Triangular kernel:
When we check the start and end points of the distributions ...
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Light tailed symmetric distribution
Is there a family of distributions that resemble the normal distribution (symmetric, spanning all real numbers, and approximately bell-shaped) but have lighter tails than normal distribution?
I'm ...
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What is the minimum of 2 Weibull distributions with the same shape parameter
If X and Y are two independent Weibull distributions with the same shape parameter, what distribution is the min(X, Y).
I am trying to find out the hazard ratio for the following case.
If I model Z=...
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Prove $Y=X$ almost surely given they have the same distribution and $Y$ is an increasing function of $X$
If $X$ and $Y$ have the same distribution and $Y=g(X)$ where $g$ is monotonically increasing, then $Y=X$ almost surely. It seems obvious, but how to prove it?
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Distribution sum of squared correlated normal random variables
Assume we have $n$ non-iid standard normal random variables $X_i$. I'm interested in the distribution of $Z=\sum X_i^2$. It is clear to me, that the sum of independent $n$ squared standard normal ...
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data preprocessing with zeros dominating
I'm working on a machine learning classification project and i faced some difficulties: all of my features distributed like this:
I'm not sure what should i do, should i use any scalers/other ...
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Parameters of the log-normal from CDF of a composition of $n$ i.i.d
Let $X_1,\ldots,X_n$ be i.i.d. log-normal random variables such that $$\log(X_i)\sim N(\mu,\sigma^2)\ \ \forall i=1,\ldots,n$$
Now let $Y$ be equal to the $\min(X_1,\ldots,X_n)$. It is quite easy to ...
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Terminology / notation for the set of variables associated with a distribution
Suppose I have a conditional (or any) distribution as so:
$$
p(A \mid B,C,D)
$$
and in the text, I want to refer to the variables $\{A,B,C,D\}$ associated with that density (or mass). Is there a ...
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Why Does the Fisher Scoring Algorithm "Work"? [duplicate]
I was reading the following link (https://en.wikipedia.org/wiki/Scoring_algorithm) on the "Fisher Scoring Algorithm". As I understand, the Fisher Scoring Algorithm is similar to the Newton-...
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How to combine two truncated distributions
We want to combine two truncated distributions to better model one phenomenon. For example, we have a Gaussian distribution, but we want to modify the right hand side tail to make it heavier. So we ...
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