# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

4,192 questions
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### Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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### 'Shrink' Step occuring often in Nelder Mead optimization?

I have an implementation of Nelder Mead, which is giving me good results. I had a bug, though, and while I managed to fix that bug, I noticed during my debugging that the 'shrink' step is occurring ...
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### Rescaling model coefficients

I have the following model $y \sim b_0 + Z \theta_0 + X \beta + \sum((X_j \circ Z) \theta_j)$ Where... $Z$ is (n, k) $X$ is (n, p) $\beta$ is (p,) $\theta$ is (p, k) $\theta_0$ is (k,) $\circ$ is ...
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### UMP test for $H_0:p=0.5$ vs $H_1:p\neq0.5$?

Let $X_1,\dots, X_n$ Bernoulli trials. I know that the UMP tests for $$H_0:p=0.5 \quad\text{vs}\quad H_1:p>0.5$$ and $$H_0:p=0.5 \quad\text{vs}\quad H_1:p<0.5$$ can be obtained with the Neyman ...
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### How to calculate median and mode from continuous data? [duplicate]

If data is continuous (not discrete) how to calculate median and mode?
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### What does conditioning on a random variable mean?

What does onditioning on a random variable mean? For example: in p(X|Y), X and Y are the random variables, so does the conditioning on Y mean Y is fixed (or non-random)?
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### Conditional expectation of two independent RV

The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $Z$ in ...
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### Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}...
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### Finding a logistic regression model which can achieve zero error on a training set training data for a binary classification problem with two features

Not sure where to begin with this question, can anyone help out?
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### Sufficient Statistic and MLE

Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
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### distribution for scaled Maximum of n independent Weibulls for $n \to \infty$

Assume that $X_1, X_2,...\sim Weibull(\lambda, k) \quad iid.$, i.e. $F(X_1\leq x) = 1-e^{-(\lambda x)^k}$ define $M_n:= \max\{X_1, ..., X_n\}$ and $\tilde{M}_n:=\frac{M_n-b_n}{a_n}$ according to ...
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### Specifying the bandwidth parameter for Local Whittle estimation

In every Local Whittle function I have seen I have to set bandwith parameter usually denoted m such that m = floor(1+T^delta). I am interested in the delta, how do I choose what value to put in the ...
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### How to compare two dispersion measures in a specific probability density function?

I am very interested in the properties of a specific probability density function (pdf) proposed in this article. It seems to me that this pdf is a very general one to capture the properties of wide ...
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### How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
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### Dependency of variable P on multiple factors, and factors interactions

In my data, I have a variable/vector P which is dependent on variables X, Y, Z, K and gender. The factors X, Y, Z are interacting. What kind of statistical tool/method I can use to explore dependency ...
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### Algorithm for modified knapsack problem

We have n-players in a game. We have a population of players we can choose from. Each player score is a normally distributed random variable and each player has a cost to add to the team. We are ...
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### Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
### AR(1) Finding $\gamma_l$
I have $\gamma_l = Cov(r_t, r_{t-l})$ as a definition in my notes and now I need to find $gamma_l$ for a series $r_t- m = p(r_{t-1} - m) + a_t$ where $r_t$ is a linear time series with expected value \$...