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Questions tagged [intuition]

Questions that seek a conceptual or non-mathematical understanding of statistics.

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Help developing intuition behind sufficient statistics (Casella & Berger) [duplicate]

Migrated from MSE I am trying to understand the following intuition for sufficient statistics in Casella & Berger (2nd edition, pg. 272): A sufficient statistic captures all of the information ...
Aaron Hendrickson's user avatar
1 vote
0 answers
29 views

Intuition behind unit roots in practice

One area where the application of unit roots to time series modelling seems very intuitive is in climate change: carbon dioxide stays in the air, so past shocks (size of flow) have a cumulative effect ...
ABCBAA's user avatar
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1 vote
0 answers
10 views

Sufficient statistic as iso-surfaces in the distribution density. Is it possible to generalise to multiple parameters?

For continuous distributions, there is a geometric intuition behind sufficient statistics that regards a multivariate probability density as several iso-surfaces. This works at least for cases where a ...
Sextus Empiricus's user avatar
3 votes
1 answer
52 views

Intuition behind relation between Gamma and Standard Normal distribution

I read if $Z$ is a random variable with a standard Normal distribution and $X=Z^2$ then $X \sim \operatorname{Gamma}(1/2, 1/2)$. I understand the math (manipulations of formulas) behind it. What about ...
Gabriele Bettineschi's user avatar
1 vote
1 answer
41 views

10 identical socks, 7 drawers: What is the probability that at least one drawer contains 3 or more socks?

My nephew asked me the question thinking I would answer him very quickly, but I got stuck until today..... There are 7 drawers and 10 socks, all the same. The socks are randomly distributed in the ...
Xavier Labouze's user avatar
4 votes
1 answer
50 views

Intuitive explanation of paradoxical interval times distribution

When I simulate a Poisson process on the interval [0,1], then the interval time between successive points follows an exponential distribution. E.g. in the code below when I select ...
Sextus Empiricus's user avatar
1 vote
1 answer
51 views

Motivation behind definition of PMF of function of $2$ variables

I am really curious to understand what motivates the definition $$p_{g(X,Y)} (g(X,Y)=z) = \sum_{(x,y)\in g^{-1}(\{z\})} p_{X,Y} (x,y)$$ where $g$ is a two variable function, and $X,Y$ are random ...
niobium's user avatar
  • 189
1 vote
0 answers
39 views

Why intuitively is standard deviation the correct thing to scale to get Central Limit Theorem?

Let me start off by saying I already know all the rigorous formulas, but let me explain why I still feel like something is missing in my understanding. There is no need for any answer going over e.g. ...
D.R's user avatar
  • 123
0 votes
0 answers
33 views

Looking for an intuitive explanation of D-Criterion for Optimal Design Problem

I know only a little about Fisher information and optimal experimental design, but I'm trying to better understand the subject. If I have an experiment composed of a single detector and my detector ...
David G.'s user avatar
  • 149
0 votes
0 answers
13 views

Question Intuition behind mathematics of activation function in a neural network.

Does this intuition behind why an activation function is used in a neural network make sense mathematically : For this example lets consider a fully connected (NOT CONVOLUTIONAL) network that ...
Stef's user avatar
  • 1
19 votes
8 answers
2k views

What is the intuition behind the idea that for linear regression, the number of observations should exceed the number of parameters?

If a population model has k independent variables and 1 intercept, why are k+1 observations required to perform OLS estimates? What is the intuition behind this?
user57623's user avatar
  • 309
8 votes
1 answer
864 views

What is a *likelihood ratio test* for a specific distribution, and how does it relate to hypothesis tests?

I'm just now being introduced to likelihood-ratio tests (LRT), and I am having trouble following the concept and terminology. For example, I posed a question about determining whether two samples {x} ...
feetwet's user avatar
  • 1,162
7 votes
2 answers
357 views

Intuition and reasoning why LASSO can only select $n$ features when $n \ll p$

I'm struggling to grasp the intuition behind why LASSO can only select at most $n$ features when $n << p$, where $n$ is the number of samples and $p$ is the number of features. I've read through ...
Jacob Bumgarner's user avatar
1 vote
1 answer
49 views

Is my interpretation of "the probability over data $X_1, ... X_N$ correct?

This may seem like a pretty simple question, but I want to make sure I am getting this right because it seems pretty foundational. I'm reading this note on conformal prediction. In the very first ...
paul's user avatar
  • 415
31 votes
15 answers
7k views

How to explain 1.5 children?

I teach undergrad stats and every year one student asks "You can't have 1.5 children" (the mean for the dataset). I am flummoxed every time to create a sensical answer. I've tried: "no ...
RLDavis's user avatar
  • 313
3 votes
1 answer
114 views

How to explain unbiasedness in basic terms?

If I take some estimator X. Lets say that X is unbiased. Suppose I have 100 samples and each sample has 5 points. I now calculate the value of X on each sample. Because X is unbiased, this means that ...
stats_noob's user avatar
3 votes
3 answers
156 views

Shouldn't we consider the difference in variance between population and a sample while calculating confidence intervals?

To comprehend the concept of confidence intervals, I came up with an example. I want to share it here for your better understanding what my question is all about. Suppose, we want to figure out what ...
Davie Blain's user avatar
14 votes
1 answer
938 views

Intuitive explanation for the fat tails of the t-distribution

Given some standard assumptions, the test statistic $$ \frac{\Delta\bar{X}}{\sigma/\sqrt{N}} $$ is normally distributed if $\sigma$ is known and t-distributed if $\sigma$ has to be estimated from the ...
monade's user avatar
  • 519
4 votes
1 answer
153 views

Markov's inequality intuitions

Can someone explain intuitively how Markov's inequality was derived? It seems plausible, but looking a it, I can't 'see' how it's true.
jbuddy_13's user avatar
  • 3,372
2 votes
0 answers
22 views

Intuition behind testing seasonality hypothesis

In this post to prove the statistical significance of a statement about a seasonality of a timeseries (every april returns are high) the author simulates alternative paths using the Monte Carlo method ...
gournge's user avatar
  • 21
0 votes
0 answers
58 views

Reference about the comparison between covariance matrices

Suppose we have 2 symmetric matrices $A$ and $B$. Then, we say that $A \succeq B$ if $A - B$ is a positive semi-definite matrix. I was wondering about the intuition and interpretation of $A \succeq B$,...
Yonatan Kurniawan's user avatar
1 vote
0 answers
75 views

what is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure? [closed]

The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $...
fromscratch's user avatar
2 votes
0 answers
64 views

Intuition behind occurence of non central chi squared distribution in conditional coordinates of a random walk

Description of background Consider a 2d random walk with drift: $$X(t) = \sum_{k=1}^t X_k \\ Y(t) = \sum_{k=1}^t Y_k$$ where each $X_k$ and $Y_k$ are independently exponentially distributed with rate ...
Sextus Empiricus's user avatar
5 votes
4 answers
674 views

Provide an intuitive example of the linearity of expectation

Can anyone explain the linearity of expectation in an intuitive way? I have been trying to understand this for far too long now. Please don't use any equations and such, try to use real world examples ...
Jeygopi's user avatar
  • 99
7 votes
1 answer
4k views

Intuitive explanation of conformal prediction

I have recently started learning about conformal prediction. I am a programmer without a strong mathematical background, but with a strong intuitive, applied background in statistics. I am trying to ...
Tripartio's user avatar
  • 2,226
2 votes
0 answers
120 views

Intuition behind rank of covariance matrix and testing hypotheses

I am trying to acquire some intuition about testing multivariate hypotheses where the test statistic involves inverse covariance matrix. As an example, suppose we have a $p$-variate random vector that ...
Richard Hardy's user avatar
3 votes
0 answers
86 views

Geometric intuition for how ridge ($L_2$) regularization helps under multicollinearity

We have some nice posts (1, 2 and likely more) illustrating multicollinearity geometrically. Now, ridge regression ($L_2$ regularization) is known to be a remedy of multicollinearity. What is the ...
Richard Hardy's user avatar
2 votes
0 answers
187 views

Understanding intuitive difference between KL divergence and Cross entropy

I know there are related questions already asked, for example this one. I also know the following: KL divergence $D_{KL}(P\Vert Q)$ is given as: $$\begin{align} D_{KL}(P\Vert Q) & = -\sum_xP(x)\...
Mahesha999's user avatar
0 votes
1 answer
65 views

Real-World Example of Correlation of Random Variables

I'm encountering a result in research that is counter-intuitive to me. Specifically, I have two matrics, $X, Y$, where $X_i$ is the ith column of matrix $X$. In my research: $\Large{\rho}$$ (\sum X_i, ...
3 votes
1 answer
335 views

Can you explain bootstrapping like I’m 5?

I think I have a handle on what bootstrapping is and why we need to use it. Please confirm if my understanding is correct: Goal of bootstrapping: To find the SE of a feature’s coefficient that you ...
Katsu's user avatar
  • 1,011
1 vote
0 answers
124 views

Concrete example of what Sufficient Statistics is [closed]

Having read articles to try to understand Sufficient Statistics. Sufficient statistics for layman A sufficient statistic summarizes all the information contained in a sample so that you would make ...
mon's user avatar
  • 1,548
0 votes
0 answers
169 views

Intuitive statistics book [duplicate]

I am looking for a statistics book that not only gives formulas or proofs but also gives intuitive explanations. For example, the standard deviation is defined by ${\sigma_x} = \sqrt{\frac{1}{n}{\sum\...
8 votes
2 answers
1k views

Is variance the area under the curve of the distribution of a population?

I am trying to understand what variance is, I already know the "official" definition "Variance is the average squared deviations from the mean" But I am trying to give it a visual ...
RodParedes's user avatar
4 votes
0 answers
397 views

What's the intuition behind the fact that sample mean and sample variance are independent when sampling from a normal population?

Let $X_1, \dotsc,X_n$ be i.i.d. from $N(\mu,\sigma^2)$, then we know that sample mean $\bar X\equiv \frac{1}{n}\sum_{i=1}^nX_i$ and $S^2=\frac{1}{n-1}(X_i-\bar X)^2$ are independent. Obviously, they ...
ExcitedSnail's user avatar
  • 2,966
3 votes
1 answer
310 views

Difference between likelihood functions for pmf vs pdf

Can someone explain the intuition behind how the likelihood function for a specific value of $\theta$ is different if $f_\theta$ is a pmf vs a pdf? I thought that it was simply the probability that a ...
Eunice Lo's user avatar
1 vote
0 answers
47 views

How to connect the intuitions to the math of adaptive processes?

Formal Definition Wikipedia gives the following definition of a process adapted to a filtration: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space; $I$ be an index set with total order $\...
Galen's user avatar
  • 9,381
3 votes
0 answers
177 views

Is there an outer product counterpart for the Covariance?

Covariance The covariance of two quantities $X$ and $Y$ within a population, $Cov(X,Y)$, is symmetric and bilinear. It is also true that $Cov(X,X) \ge 0$. So, clearly $Cov(X,Y)$ qualifies as an inner ...
Fermion's user avatar
  • 31
2 votes
1 answer
37 views

Linear models when responses have no link

I am studying normal linear regression and wanted to ask a question about its utility when working with independent RV. Suppose that we have for $k \in [1,\dots,n]$, $$Y_k = \beta_0 + \beta_1x_{k1} + \...
Kilkik's user avatar
  • 445
0 votes
0 answers
62 views

Intuition on expected value of an estimator [duplicate]

When we don't know what's the mean of a normal distribution we try to estimate it and after a time we get lucky and have the true mean (in a magical way). What does it mean the expected value of the ...
Abderrahmen Hamdi's user avatar
1 vote
0 answers
42 views

Interpretation of covariance and linear dependency [duplicate]

What is the best interpretation of covariance you can give ? I know that if $X$ and $Y$ are random variables, then if $Cov(X,Y)>0$, then if realizations of $X$ are higher than expected, then ...
Kilkik's user avatar
  • 445
3 votes
1 answer
140 views

Mean of geometric distribution is odds?

Context: I mean the $P(X=k)=(1-p)^k p$ not the $P(Y=k)=(1-p)^{k-1} p$. Apparently the mean of the 1st kind of geometric is $\frac{1-p}{p}$ instead of $\frac{1}{p}$ for the 2nd kind of geometric. I ...
BCLC's user avatar
  • 2,444
3 votes
1 answer
143 views

What does conditional independence mean semantically?

I've just spent the last 3 hours reading every post, question, Medium article, and textbook entry on conditional independence, and I still don't really understand it. Can somebody explain what it ...
NaiveBae's user avatar
  • 257
0 votes
0 answers
390 views

What exactly does the Box-Cox transformation do to a time series?

If I were to try and rephrase the argument in the original Box-Cox paper in my own words, I would say something like the following: given a model $$ y = x \beta , $$ if the residuals do not appear to ...
Anthony's user avatar
  • 540
2 votes
1 answer
527 views

Why do we need a smaller sample size to detect a smaller proportion?

The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions: This assumes an infinite population size, and the confidence intervals are fixed ...
Mhairi McNeill's user avatar
1 vote
0 answers
588 views

Why PCA is invariant under rotation

Lets say that we have a matrix of variables (the columns are variables and rows are the observations) called X whenre X = [x1, x2, ...., xp] where ...
Amir Jalilifard's user avatar
1 vote
0 answers
249 views

Does the number of samples, as opposed to the sample size in each sample, matter for the Central Limit Theorem? [closed]

(1)So here is a formula that describes CLT I found at https://en.wikipedia.org/wiki/Central_limit_theorem. According to the first part of the explanation, n as in Xn describes the number of samples(i....
e1phy's user avatar
  • 11
1 vote
0 answers
83 views

Why is my intuition about probability in this regard so flawed? [closed]

Take the following example: Take a sample from 100 people and measure their height. Assume that we know that height is approximately normally distributed, with a sample mean of 175 cm and sample ...
Sen90's user avatar
  • 111
13 votes
5 answers
2k views

What's complicated about regression to the mean?

Note: I am a bit of a novice when it comes to statistics and data analysis. Reading the chapter on regression to the mean in Kahneman's Thinking Fast and Slow, I came across the following passage: ...
ciru_4011's user avatar
  • 133
2 votes
0 answers
169 views

Intuition behind log in kl distance

So, let's start stating that I already read both Why KL-Divergence uses "ln" in its formula? and What is the role of the logarithm in Shannon's entropy? ... However, I still have no ...
Alberto's user avatar
  • 1,217
1 vote
0 answers
294 views

Intuition about the relation between joint distribution, marginal distribution, and conditional distribution

The wording "intuition" might be a bit imprecise. I want to discuss how we visualize in our head going from one to another among the joint PDF, marginal PDF, and conditional PDF. To make the ...
whoknowsnot's user avatar

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