All Questions
Tagged with variance distributions
198 questions
110
votes
13
answers
74k
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 ...
- 14.9k
69
votes
9
answers
58k
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 ...
41
votes
4
answers
36k
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 ...
- 513
37
votes
2
answers
8k
views
Distributions other than the normal where mean and variance are independent
I was wondering if there are any distributions besides the normal where the mean and variance are independent of each other (or in other words, where the variance is not a function of the mean).
- 17.9k
18
votes
2
answers
43k
views
Var(X) is known, how to calculate Var(1/X)?
If I have only $\mathrm{Var}(X)$, how can I calculate $\mathrm{Var}(\frac{1}{X})$?
I do not have any information about the distribution of $X$, so I cannot use transformation, or any other methods ...
- 690
14
votes
4
answers
6k
views
Meaning of "Overdispersion" in Statistics
I am trying to understand what "overdispersion" means in statistics.
Based on the Wikipedia page, "overdispersion" is defined as follows : "In statistics, overdispersion is ...
14
votes
3
answers
11k
views
What criteria must be met in order to conclude a 'ceiling effect' is occurring?
According to The SAGE Encyclopedia of Social Science Research Methods…
[a] ceiling effect occurs when a measure possesses a distinct upper
limit for potential responses and a large concentration ...
- 141
12
votes
2
answers
8k
views
How to parameterize the ratio of two normally distributed variables, or the inverse of one?
Problem:
I am parameterizing distributions for use as a priors and data in a Bayesian meta-analysis. The data are provided in the literature as summary statistics, almost exclusively assumed to be ...
- 7,270
11
votes
3
answers
10k
views
Variance of a distribution of multi-level categorical data
I am currently analyzing large data sets with various characteristics (such as city). I wanted to find a measure which would essentially say how much or how little of a variance there was across the ...
- 213
11
votes
2
answers
738
views
Independence of Mean and Variance of Discrete Uniform Distributions
In the comments below a post of mine, Glen_b and I were discussing how discrete distributions necessarily have dependent mean and variance.
For a normal distribution it makes sense. If I tell you $\...
- 67.1k
10
votes
4
answers
2k
views
Inverse function of variance
For a given constant number $r$ (e.g. 4), is it possible to find a probability distribution for $X$, so that we have $\mathrm{Var}(X)=r$?
- 231
10
votes
1
answer
17k
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if 2 random variables have exactly same mean and variance [duplicate]
If two continuous random variables have exactly the same expected value and variance, do they always have the same distribution?
- 103
10
votes
2
answers
21k
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What are the error distribution and link functions of a model family in R?
When building models with the glm function in R, one needs to specify the family. A family specifies an error distribution (or variance) function and a link ...
- 203
10
votes
3
answers
2k
views
How to test whether the variance of two distributions is different if the distributions are not normal
I'm studying two geographically-isolated populations of the same species. Inspecting the distributions, I see that both are bimodal (there's some seasonality to their occurrence), but the peaks in one ...
- 337
10
votes
1
answer
4k
views
Variance of Normal Order Statistics
Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics.
Are there any references ...
- 1,402
9
votes
1
answer
20k
views
Distribution of sum of squares of normals that have mean zero but not variance one?
I am trying to find the distribution of a random variable that is calculated according to $Y:=\sum_{i=1}^n X_i^2$ where $X_i $ is distributed as $ \mathcal{N}(0,\sigma^2_i)$. Does there exist a ...
- 1,007
9
votes
2
answers
39k
views
Expected value and variance of the square root of a random variable
Let $X$ be a univariate continuous random variable for which I can calculate all raw and central moments.
Is there an exact way to calculate $E[\,\sqrt{X}\,]$ and $\mathrm{Var}[\,\sqrt{X}\,]$ in this ...
- 789
8
votes
3
answers
1k
views
Looking for a distribution where: Mean=0, variance is variable, Skew=0 and kurtosis is variable
I am aiming to run simulations in order to estimate the influence of the distribution of $Y$ (independent variable) on a certain binary outcome $X$ (dependent variable). $Y$ must always has a mean of ...
- 5,182
8
votes
1
answer
7k
views
What is the distribution of the sample variance for a Poisson random variable?
The mean and variance of a Poisson random variable $X$ are both $\lambda$ but what is the distribution of the $\operatorname{var} X$ across a series of experiments recalculating each time? I would ...
- 363
8
votes
1
answer
5k
views
Something like Central Limit Theorem for variance and maybe even for covariance?
CLT states in short, that sum/mean of random iid variables from almost any distribution approaches normal distribution.
I failed to find information about asymptotic behavior of sample variance when ...
- 2,200
7
votes
1
answer
138k
views
Var(XY), if X and Y are independent random variables [duplicate]
if X and Y are independent Random variable then what is the variance of XY?
- 71
7
votes
1
answer
11k
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When does a distribution not have a mean or a variance? [duplicate]
I believe I read today a phrase which went something like this:
If a distribution has a mean and a variance ...
So I guess that means some distributions do not have means or variances?
I fiend ...
- 2,055
6
votes
3
answers
2k
views
Binomial distribution intituition for N
I am unable to convince myself intuitively as to why the variance of a binomial distribution increases with increase in n (number of trials). In general, I expect that as n increases, the distribution ...
- 113
6
votes
2
answers
46k
views
Standard error from correlation coefficient
Many studies only report the relationship between two variables (e.g. linear or logistic equation), $n$, and $r^2$. I want to use these reported statistics to reproduce this relationship with its ...
- 61
6
votes
2
answers
1k
views
Variance of the modulus of a random variable
Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. What is the upper-bound on the variance of $Y=\left|X\right|$?
My gut feeling says that $\operatorname{Var}(Y) \leq \operatorname{...
- 591
6
votes
1
answer
2k
views
Variance of the Poisson Binomial Distribution
Consider a sequence of $n$ independent Bernoulli trials drawn from a list of biases $p_1,p_2,...,p_n\in[0,1]$, respectively. We set the random variable $X$ to be the sum of these trials. On wikipedia, ...
- 427
6
votes
1
answer
458
views
How to estimate variance of sample variance?
Given an arbitrary sample, sample variance would be calculated. But how the variance of sample variance should be estimated? I tried to do some simulations using influence functions estimation methods....
- 211
5
votes
3
answers
6k
views
Can IQR ever be larger than standard deviation?
As I understand it the IQR specifies the dispersion of the data by taking the difference between Q3 and Q1 (i.e., the range that the middle 50% of the data lies), while std specifies dispersion using ...
- 337
5
votes
3
answers
309
views
Mean absolute difference for the gamma distribution
A wikipedia entry states that the mean absolute difference for the $\Gamma(k,\theta)$ distribution is $k\theta(4I_{0.5}(k+1,k)-2)$ where $I_z(x,y)$ is the regularized incomplete beta function, equal ...
- 576
5
votes
1
answer
1k
views
Variance of unbiased estimator for the shape parameter of Pareto distribution
I'm interested in getting the error bounds of the unbiased estimator of the shape parameter ($\alpha$) using maximum likelihood method of Pareto distribution.
The unbiased estimator is known to be
...
5
votes
1
answer
477
views
Variance of $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$
Here the $X_i$'s are i.i.d. and such that convergence in distribution for the infinite sum, is guaranteed. Probably the easiest case is when $X_i$ has a Bernouilli($p$) distribution, then $Z$ has a ...
5
votes
1
answer
541
views
Expected value and variance of moving a token on a cartesian plane based dice rolls
A fair four-sided die has its sides labeled U, D, L, and R, respectively. A token is placed at (0, 0) on the Cartesian plane and the die is then rolled repeatedly. After each roll, the token is moved ...
- 55
5
votes
1
answer
3k
views
How do we derive that $S^2$ is chi-squared distributed (with $n-1$ df)?
The claim is that $$(n-1)S^2/\sigma^2$$ is chi squared distributed with degrees of freedom $n-1$.
$(n-1)S^2/\sigma^2$ can be written as $$\sum_i^n \left(\frac {x_i-\mu}{\sigma}\right)^2-\left(\frac {...
- 2,987
5
votes
2
answers
1k
views
Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation?
Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation?
I've only taken AP statistics, but I would like to ...
- 51
5
votes
1
answer
4k
views
What is the difference between "scale parameter" and the variance?
I would like to understand the difference between "scale parameter" and the variance of a distribution? I found, that the "scale parameter", scaling the width of a distribution is mentioned when ...
- 123
5
votes
1
answer
2k
views
distribution of sample variance of correlated observations
It is well known that if we have n i.i.d. observations of a normal random variable, then Cochran's theorem tells us that:
$\frac{(n-1)s^2}{\sigma^2} \widetilde{} χ^2_{n-1}$
But what if the samples ...
- 449
5
votes
1
answer
6k
views
What is the expectation of exponential of the product of two random variables?
I am looking for examples of probability distributions that would allow me to characterize the distribution (at least approximately) and to compute the first two moments exactly of:
$$
e^{aXY}
$$
...
- 157
4
votes
2
answers
201
views
Distribution of a random segment on a string
I have a linear string of unit length, and I randomly sample two locations a and b from Uniform(0, 1). Then I cut the string at these two locations to get a sub-string. What is the distribution for ...
- 165
4
votes
1
answer
1k
views
Is it possible for a distribution to have infinite variance but finite covariance or vice versa?
Is it possible to have distributions s.t. one/both have infinite variance, but finite covariance? What about finite variance but infinite covariance?
If so, what are example distributions/what is the ...
- 55
4
votes
1
answer
186
views
equal *population* variances in paired t test
I I want to perform a paired t-test to check if there's some effect, I have the distribution of "before" and the distribution of "after" the manipulation. Do I need to assume the ...
- 41
4
votes
2
answers
188
views
If $|E(X)|< 1$ and $E(X^2)<1$, can we have $1 - E(X^2) = (1 - E(X))^2$?
Of course $X=0$ works, but I am looking for a non-singular solution. I haven't made much progress to solve this problem. However, let $\mu_2 = E(X^2)$ and $\mu_1 = E(X)$. For the equality to hold, we ...
4
votes
1
answer
3k
views
One sample test of uniformity in R
I have a dataset of two columns: one with IDs and one with a column of single digits (0-9) (see below). I would like a statistical significance test for whether the data is uniform. Ideally, I would ...
- 139
4
votes
2
answers
72
views
How $Var[e^{\frac{-1}{X+a}}]$ varies with $n$ where $X \sim Bin(n,p)$?
I have a binomial random variable $X \sim Bin(n,p)$. I am interested in the variance of a function $f(X)$ given by :
$f(X)=e^{\frac{-1}{X+a}}$. Here $a>0$.
Specifically, I would like to know how $...
- 224
4
votes
1
answer
303
views
Variance of beta distribution (fastest way)
Suppose a Random variable $X \sim \mathrm{Beta}(a,b)$
Find the $\mathrm{Var}( \frac{X}{1-X} ) $
My initial approach is to calculate $\mathrm{E}( \frac{X}{1-X} ) $ and $\mathrm{E}( [\frac{X}{1-X}]^...
- 213
4
votes
1
answer
685
views
What is the maximum entropy distribution given the median (instead of the mean)?
Given that the median seems to be a more robust statistic than the mean/average, I was wondering if there is a solution of the maximum entropy distribution given the median (or the median and some ...
4
votes
2
answers
913
views
Moments of truncated Student's $t$-distribution
I performed random sampling on a Student's $t$-distribution. I used SciPy to calibrate my parameters and then truncated my allowable values to the maximum and minimum observation in the data for ...
- 171
4
votes
1
answer
2k
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Why will the validation set error underestimate the generalisation error?
In my book about machine learning the concept of a validation set is introduced. It's a subset of the training set that is used to "train" the hyperparameters. More specifically, the validation set is ...
- 395
4
votes
1
answer
478
views
Hierarchical Bayesian Regression, Can an Inverse-Gamma distributed Variance look Normal or t?
Using Peter Hoff's book, A First Course in Bayesian Statistical Methods, I used some of my own data to fit a Hierarchical Bayesian Regression following his example. In his book, he utilized a Gibbs ...
- 619
4
votes
2
answers
4k
views
Asymptotic distribution of MLE (log-normal)
Say we have a sample $X_{1},...,X_{n}$ from a log-normal distribution with parameters $\mu$ and $\sigma^{2}$. That is, $\ln(X)$~$N(\mu,\sigma^{2})$. Let $T_{n},Z_{n}$ denote the MLE's for $\mathbb{E}(...
- 41
4
votes
1
answer
2k
views
What is the correct way to add and subtract skewness from a distribution?
If I recall correctly, we can add and subtract variance if variables are independent, and have a mean of 0.
I have two distributions that are summed up: (a) one with high variance, low skewness and ...
- 143