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Here's a straightforward one. Consider a uniform population with unknown upper bound $$X \sim U(0, \theta)$$ A simple estimator of $\theta$ is the sample maximum $$\hat \theta = \max(x_1, x_2, \ldots, x_n)$$ This is a biased estimator. With a little math you can show that $$E[\hat \theta] = \frac{n}{n+1} \theta$$ Which is a little smaller than $... 6 A very commonly used consistent but biased estimator used is that of the estimated standard deviation. If we are looking at a simple situation in our data is distributed as$x_i \sim N(\mu, \sigma^2)$, then sometimes the MLE estimate of$\sigma$is used, ie$\hat \sigma^2 = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar x)^2$This is, of course, a biased but ... 4 Not shure, whether this is worth an answer or just a comment, but I want the room this forum gives for answers only. Have look at this golf putting example, that Andrew Gelman gave in a webinar about stan. Forget about the Bayes-aspect of it. It just shows a standard model compared with an informed model and how the result improves when knowledge about the ... 3 This is known as "data dredging". You can read wikipedia. The simplest case is when testing 1000 independent hypothesis each with type I risk 1%, you can be almost use one of them will be positive, as a matter of chance. 3 For realistic and educational purpose, I may suggest to use Boston data set. It is on UCI repository, and popular in both statistics and machine learning community. In addition, it has "reasonable" amount of rows and columns (~500 observations, and 14 variables). In R, it is included in MASS package. A staring point can be using housing age and value to do ... 2 I found working out an ICA problem (a simple one about unmixing audios) for the Stanford cs229 course very helpful in understanding its inner working. The basics of ICA isn't that complicated. Check these out: Andrew Ng's note on ICA: http://cs229.stanford.edu/notes/cs229-notes11.pdf The simple problem on unmixing audio: it's the 4th question. My ... 2 On "Deep Learning with Python" by F. Chollet (https://www.manning.com/books/deep-learning-with-python) he uses the "Boston Housing Price" dataset for an introductory example in regression. In Python you can easily import the data using: from keras.datasets import boston_housing 1 I have observations$\{3,2,1,0\}, n=4$I count$ \overline{X}= \frac{3}{2} \hat{γ_1}(0)=\frac{5}{4}, \hat{γ_1}(3)=-\frac{9}{4}$then$ \widehat {Var}(X_1 + X_4)=\widehat {Var}(X_1)+\widehat {Var}(X_4)+2\widehat {Cov}(X_1, X_4)= 2\hat{γ_1}(0)+2\hat{γ_1}(3)= \frac{5}{2} - \frac{9}{2} = -2<0$1 ...the classification into "high", "medium" and "low" correlation seems rather vague to me. You are right - the notion of correlation as "high", "medium" or "low" is a fuzzy concept. Vagueness of the categorisation is an inherent aspect of what happens when you impose discrete categories over a continuum to create a concept of this kind. It involves a ... 1 So, your understanding is essentially right. So simply apply your understanding in the real world to something measurable. For example, what is something that, when larger measurements are obtained on one variable are accompanied by larger measurements on another variable? Do you think weight and height would be highly correlated? I think so as taller ... 1 But missing is seemingly why we might choose the Poisson distribution over other choices This is indeed a frequent trait of probability textbooks – and even of research works in statistics. Somewhat of a taboo it seems. It was pointed out by the statistician A. P. Dawid (1982, § 4, p. 220): Where do probability models come from? To judge by the ... 1 What is meant by "torture" is ambiguous. However, I believe that subjecting data to procedures for which it is not intended is a form of this torture. Anscombe's quartet is a classic example of this, subjecting four sets of data to linear regression when three of them clearly do not fit the assumptions. Credit A second kind of data torture is ... 1 I prefer the term "cherry picking", which may result from testing tons of hypotheses using unadjusted alpha-level (same as @Benoit Sanchez's). 1 So I have wandered online and found some examples of negative autocorrelation : If you've ever seen a row of cabbages growing in a garden, you'll frequently notice an alternating pattern--big cabbage, little cabbage, big cabbage, little cabbage, etc. This happens because one cabbage might have a slight edge in growth. It extends into its neighbor's space, ... 1 Here is such an example. I will describe the process in terms of its sample paths. It's a simple process, it only has four sample paths:$\omega_1$,$\omega_2$,$\omega_3$,$\omega_4$. This means that the only possible outcomes are: $$...,\ X(-1) = \omega_i(-1),\ X(0) = \omega_i(0),\ X(1) = \omega_i(1),\ ...$$ One outcome for every$i \in \{1,2,3,4\}\$. ...