# 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 ...
1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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?
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### 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} ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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### 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.
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### 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 ...
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### 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 $A \succeq B$,...
1 vote
75 views

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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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 ...
1 vote
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....
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### 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 ...
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### 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: ...
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