Linked Questions

3
votes
1answer
8k views

What is the difference between Mean Squared Deviation and Variance?

I am doing some tutoring for an AS-Level maths student and unfortunately for me they are doing statistics. This is not my strong point, mainly from the point of view of remembering all of the ...
4
votes
1answer
6k views

Estimating the covariance matrix in linear discriminant analysis

I was trying to derive the equations from page 109 in "elements of statistical learning" (image below) To be honest, I am not sure how the covariance $\Sigma$ is estimated (the third bullet point in ...
0
votes
3answers
7k views

Measure the fluctuation of data

I have 2 series of data, for example: s1: 0.3 0.3 0.4 0.8 0.6 0.5 0.7 s2: 0.7 0.7 0.6 0.8 0.7 0.5 0.6 It's easy to see that ...
6
votes
2answers
162 views

What value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal?

I know that $\bar{x}$ makes absolute result of $\sum_{i=0}^n (x_i-\alpha)$ minimum. In fact it makes it zero. But how to find what value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal? What ...
2
votes
2answers
2k views

Expected number of successes from $N$ Bernoulli trials with different $p$

Suppose I have N probabilities $(p_1, p_2,...,p_N)$ that represent the chance that each that a corresponding test was passed. How do I apply the Bernoulli distribution to determine the expected number ...
11
votes
2answers
1k views

Why does Restricted maximum likelihood yield a better (unbiased) estimate of the variance?

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
4
votes
1answer
225 views

An unbiased estimator of σ³

As it was suggested in the linked answer, $s_n = \sqrt{\frac{\sum_{i = 1}^n (x_i - \bar{x})^2}{n - 1}}$ is not an unbiased estimator of $\sigma$. I suspect neither $s_n^3 = \sqrt{\frac{\sum_{i = 1}^n ...
1
vote
1answer
1k views

Covariance Matrix vs. Pairwise Covariance Matrix?

I found this equation here to calculate a covariance matrix of any number of variables using matrix algebra. $$\frac1{N} (X - 1\bar{x})^T(X - 1\bar{x}^T) $$ For a given matrix $X$ with $N$ samples. ...
4
votes
2answers
214 views

Sample Variance and Dividing by $n-1$

In this video... https://www.youtube.com/watch?v=sHRBg6BhKjI ...and in many others, the explanation for why when calculating the sample variance we divide by $n-1$ instead of by $n$ is the following:...
1
vote
1answer
850 views

Variance estimator of Bernoulli RV (in CI)

Assume n samples from Bernoulli distribution with unknown parameter p, i. e. $x_{1},....,x_{n}\underset{iid}{\sim}Ber\left(p\right)$ It is known that the confidence interval is given by: $CI=\hat{...
3
votes
1answer
497 views

Dividing by degrees of freedom [duplicate]

When estimating parameters such as (I don't care about this specific instance particularly) Variance of a random variable X, one usually adopts Bessel's correction, i.e. using the formula $\hat{Var}{(...
1
vote
0answers
650 views

Estimation of variance: How to bring Bessel's correction together with degrees of freedom?

I have been considering multiple textbooks to find out the reason that the denominator of the estimation of the population variance is n-1 rather than n. Depending on the book, two reasons are given: ...
3
votes
1answer
138 views

Why is $\frac{\sum^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$chi-square distributed with $n-1$ degrees of freedom?

On Wikipedia, it says If $X_{i};i=1,\ldots ,n$ are independent normal $(\mu ,\sigma ^{2})$ random variables, the statistic $$\frac{\sum\limits^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$$ ...
0
votes
3answers
159 views

Variance (different definitions)

I'm new to statistics and reading stats books I found different definitions for Variance. Definition1: $ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 $ Definition2: $ s^2 = \frac{1}{n} \sum_{i=...
5
votes
2answers
132 views

Definition of a sample: can it include the same object twice?

Wikipedia defines a sample as: a subset of a population. While exploring the reason why we divide by $(n-1)$ instead of $n$ when calculating standard deviation (discussed in this question), I came ...

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