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Bounds on the conditional variance of a truncated binomial

I apologize, but I need to change the notation a little to make it a little more consistent with stats references and easier to follow. I'll switch to your notation at the end. $$X \sim binomial(n, p)...
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2 votes

What is the impact of duplicate data on the variance of regression coefficient?

The coefficients themselves will no change. Imagine you perform the analysis on the first dataset, and plot the regression line with the datapoints around the regression line. Now what would happen if ...
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  • 570
1 vote

What is meaning of such notations in general?

$$\mathbb{E}_{(\mathbf{u}, \mathbf{\sigma}, \mathbf{Y}_0)\sim N(0,1)\otimes \mu_D\otimes N(0,1)}\left[ e^{\mathbf{\sigma} t}\mathbf{Y}_0\mathbf{u}\right]=\int\int\int e^{\mathbf{\sigma} t}\mathbf{Y}_0\...
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4 votes

Understanding the concept of "Bounded in probability"

We may depict the relevant distributional properties of any sequence of random variables by plotting the distribution functions of their absolute values. Such functions have values between $0$ and $1,...
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2 votes

In elastic net regularisation, will dividing the OLS term the number of observations cause misleading results when cross-validating?

It's correct that when the sample size is fixed, there is not a difference between the two statements of the optimization problem. Your demonstration in the revised question makes it clear that the ...
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0 votes

Error Terms in the ANOVA Summary Table

Partial answer based on my Comment: For example, in your output. The 'Error Term' of the F-ratio for testing effect A is $\mathrm{AS(BC)},$ so the F-ratio is $\mathrm{MS(A)} = 693\,600$ divided by $\...
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2 votes

Proof of statistical assumptions behind the methods or tests

A lot of older applied statistics and econometrics textbooks start with a wide range of assumptions, without always making them clear. Readers then learn a wide range of tools based on (possibly ...
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2 votes

Proof of statistical assumptions behind the methods or tests

Understanding derivations (and how the assumptions are used in those derivations) is definitely helpful when using statistics. It can also make it a lot easier when you are trying to figure out how ...
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0 votes

football probability

How can I find the probability of the best player score the first goal? This is given in the question "given that T scores a goal, the probability that the best player of the team scored it is ...
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0 votes

How can I calculate Posterior Distribution, analytically with given information?

For $ x=1 $ $ \dfrac{p(\theta)p(x=1 | \theta)}{p(\theta_1)p(x=1 | \theta) + p(\theta_2)p(x=0 | \theta_2)} = \dfrac{p(0)p(x=1 | 0)}{p(0)p(x=1|0)+p(1)p(x=1|1)} = \dfrac{0.1665}{0.417} = 0.399280576 \...
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0 votes

If the level of a test is decreased, would the power of the test be expected to increase?

Ceteris paribus, when you decrease the significance level $\alpha$ in a classical hypothesis test, you are increasing the amount of evidence required to reject the null hypothesis. This means that ...
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2 votes

Conditioning of join gaussian over a line

Let $X$ and $Y$ be jointly normal random variables with means $\mu_X, \mu_Y$, and covariance matrix $\Sigma$. (We do not need that $X$ and $Y$ are independent, although it does simplify some ...
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2 votes

Conditioning of join gaussian over a line

A bivariate normal density can be likened to a piece of bologna (or did I mean to write baloney?) about which Americans often say "No matter how you slice it, it is still bologna". The ...
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1 vote

Mathematic modelling vs. machine learning

Which one is superior? Neither is superior to the other. There are cases where one is preferable to another. I will take what you mean by "mathematical modelling" to mean differential ...
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0 votes

Mathematic modelling vs. machine learning

Mathematical modelling appears in a lot different fields, especially statistics that are just a way of making models using empirical data or introduce randomness. The is no "superiority", ...
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3 votes

Why do discrete choice models (such as MNL) not require test set?

Multinomial logistic regression can and often does consider out-of-sample performance. For instance, LeCun (1998) applied multinomial logistic regression to the pixels of the MNIST handwritten digits, ...
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0 votes

sufficient, minimal, complete

Consider a Normal $\mathcal N(\mu,1)$ sample, $x_1,\ldots,x_{2n}$. Then both $\bar X_{1:n}$ and $\bar X_{(n+1):2n}$ are complete, insufficient, and not a function of one another.
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5 votes

variance of the difference of two independent variables is the sum of variances

The relevant equation is $V(X-Y) = V(X)+V(Y)-2Cov(X,Y).$ The correlation may seem small to you, but @whuber's Comment is exactly correct. Computations in R for your apple prices: ...
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1 vote

Are there possibilities to determine 95% confidence interval for right skewed data?

Usually integer data, like days, is modeled using a Poisson (or Negative Binomial) regression model, both instances of what is called a generalized linear model (GLM). $$Y \sim Poisson(\lambda)$$ $$...
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2 votes
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two sample t-test for non-normal population

The same logic (application of CLT and Slutsky's theorem) will also show convergence of the Welch T-test in the same way as the equal variance T-test. In both cases the denominator is a consistent ...
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2 votes
Accepted

Proof that a necessary condition for characteristic roots to lie inside unit circle is $\sum\limits_{i=1}^{n} a_{1} < 1$

Let $$P(z) = z^n - \sum_{i=0}^{n-1} a_i z^i.$$ Notice that $$P(1) = 1^n - \sum_{i=0}^{n-1} a_i 1^i = 1 - \sum_{i=0}^{n-1} a_i.$$ If all the roots are inside the unit circle, there are no roots on the ...
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1 vote

Estimate the value of a sigmoid function over expectation

No unbiased estimator exists, when $p(x)$ is Categorical distribution: Unbiased estimator of exponential of measure of a set? For the binomial distribution, why does no unbiased estimator exist for $1/...
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1 vote
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Suppose $X$ and $Y$ have joint pdf $f(x,y) = 0.5,$ for $0 \le x \le 1;$ $0\le y\le2.$ Evaluate $P (Y >X).$

Graphical comment: Maybe this will get you started. First, $f(x,y)$ is the joint density function of $X$ and $Y.$ Your distribution is uniform on the rectangle shown below. $10\,000$ realizations of $(...
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2 votes
Accepted

Model performance when ground truth is not available

I am not sure I understand your first technique, because, AFAIK, the reconstruction error of autoencoders is what is used as the score for anomaly detection in the first place, so your first technique ...
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0 votes

How to test for an "interaction" between two interactions

Your sketches of hypothetical combinations of the 3 binary predictors seem to show 8 different outcome values. That would require a 3-way interaction among those 3 individual predictors to give 8 ...
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0 votes

Converting scaled parameters to unscaled parameters in exponential regression

Update: I believe I figured out a solution for model 1, by using a natural log transformation to make the model linear. It seems to work, at least if I change the additive error term to a ...
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1 vote

Are financial asset prices (Pt) log-normally distributed or arithmetic returns (Pt/P0)?

I am assuming you are referring to stocks as financial assets instead of derivatives. Sometimes, stock price behavior is said to follow geometric Brownian motion. You can look up if unfamiliar. If you ...
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3 votes

Partials of PDF with no closed form solution

Given your description of your problem, I'm going to assume that your characteristic function is in a closed form that is simple to differentiate, but your density isn't. I'm also going to assume ...
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1 vote
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Infinitesimal Robustness, influence function of $T$ at $F$

You are working in a function space, specifically, the space of probability distributions equipped with some useful topology. In this case, we can choose two probability distributions in that space - ...
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2 votes
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using logsumexp in softmax

The idea of working in log-space to avoid underflow requires that the intermediate objects you use to track progress are themselves on the log-scale --- you only convert back to regular scale at the ...
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1 vote

using logsumexp in softmax

matrix is on the log scale. If matrix were not on the log scale, then you would only want to do ...
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1 vote

How to plot the random and best model in Lift and gain charts?

I have never seen these kinds of plots before, but here's the idea. The "random" line is representing the results you would get if you did completely random guessing - so this is the "...
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3 votes
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How do you get P(A|C) from P(A, B|C)?

Ignore the conditioning on $C$ for a moment. How does one in general get a marginal distribution $P(A)$ from a joint distribution $P(A,B)$? One "integrates out" the dependence on $P(B)$, ...
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-1 votes

What is meant by a "random variable"?

The term “random variable” is confusing enough, the better term would be “a varying value”. A particular object - for example an apple - has in a particular time a fixed value, e.g. its mass is 136 ...
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1 vote
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Is this Bayes Theorem?

Yes, it's just Bayes theorem with some additional probability rules applied (and then some additional manipulations). e.g. one such rule being used is $p(z,\theta) = p(z|\theta)\, p(\theta)$, which ...
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0 votes
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Why can de-noising diffusion models be sampled with Gaussian distributions?

We can compute the distribution of the noisy sample after $t$ iterations of the forward process in closed form, using equation 4 in the paper: $$q(\mathbf{x}_t|\mathbf{x}_0) = \mathcal N(\mathbf{x}_t|\...
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0 votes

Compare Paired Data from 2 Timeframes of Different Lengths

I would suggest learning about generalized linear mixed effects models (there is much more to be learned than can be covered in a forum like this, take a class or at least an online tutorial). For ...
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2 votes

Maximum Likelihood Estimation for data with non normal distribution

Maximum Likelihood Estimation is about finding the parameters that maximize the likelihood of a probability model that you assume the data is generated from. So if your model was $$ y = \alpha + \beta'...
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  • 271
1 vote
Accepted

Convert from log-normal distribution to normal distribution

If $X \sim \mathrm{N}(\mu, \sigma^2)$ and $Y=e^X$, then $Y \sim \mathrm{Lognormal}(\mu, \sigma^2)$. The random variable $Y$ has mean $m=\mathbb{E}(Y)=\exp(\mu+\sigma^2/2)$ and variance $v=\mathrm{Var}(...
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18 votes
Accepted

Suppose I have 100 integers and I sample 10 without repetition. What is the expected rank of the lowest out of 10 samples?

To obtain an answer we must know how many ties there are among these $100$ integers and where they occur: that's too complicated and likely is not the intent of the question. (Nevertheless, the ...
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