Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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2
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2answers
49 views

Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
0
votes
1answer
17 views

How many time the standard deviation, do I need to travel from mean in both directions such that I cover a given percentage of data?

I do not have much experience in Statistics. However, I read this rule on a page and followed it up on Wikipedia: https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule I wanted to know ...
0
votes
2answers
22 views

Show that $(\bar{X})^2$ is not an unbiased estimator for $\mu^2$

If $X_1, ... , X_n$ are n identical distributed independent random variables each with mean $\mu$ and variance 1. A little confused by this question. Is it asking for if $(\bar{X})^2$ != $\mu^2$. ...
0
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0answers
8 views
+50

Transforming a categorical distribution by repeating trials and taking a plurality

Suppose you have a K-sided, weighted die. This is represented by a categorical distribution. Now, let's say you roll the die N times, and then pick a "winner" by choosing whichever outcome has a ...
1
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2answers
19 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
1
vote
0answers
38 views

probability of rank of a number

Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments. First experiment: I pick one of the ...
3
votes
2answers
49 views
+50

Reference Request - Statistics Book with exercises

I'm looking for an as complete as possible statistics book with exercises, including the following topics: Probability Review Random Variables and Samples Descriptive Statistics Estimation (...
2
votes
1answer
748 views

Two-tailed hypothesis test; Why do we multiply p-value by two?

I understand that in a two-tailed hypothesis test, we must multiply the p-value by two. i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed ...
1
vote
0answers
31 views

Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
0
votes
0answers
29 views

expectation and variance of an implicit estimator

Suppose the following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+...
18
votes
1answer
60k views

Sample Standard Deviation vs. Population Standard Deviation

I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values: Sample Population I ...
0
votes
0answers
38 views

Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
2
votes
1answer
384 views

Probability of a slot having exactly $K$ elements

From this question asked in an interview: Consider a hash table with $M$ slots. Suppose hash value is uniformly distributed between $1$ to $M$. Suppose we put $N$ keys into this $M$-slotted ...
2
votes
1answer
29 views

sequential anova r

I am a really confused. Assume we have a multiple regression model: $$ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 +\cdots+ \beta_k x_k $$ Using R we can make a test: $$ H0: \beta_1 = \beta_2 = \...
1
vote
1answer
332 views

Show that the LS estimator b is unbiased for $\beta$ when regressing without intercept

Okay so I have gotten down to $$b=\beta + \frac{\sum_{i=1}^{n}x_i \varepsilon_i}{\sum_{i=1}^{n} x_i^2}$$ but I cannot figure out how to show that second term is $0$.
0
votes
0answers
15 views

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$

Show $G^2=2\sum o \log \frac{o}{e}$ is approximately $X^2=\sum \frac {(o-e)^2}{e}$ $o_i$ = observed $e_i$=expected (I removed $i$'s for ease) The solution is: $$G^2=2\sum o \log \frac{o}{e}$$ $$=2\...
0
votes
0answers
17 views

Convolution of multiple exponential distributions

I'm trying to figure out the derivation presented on page 442 of this paper. Given a probability distribution $$f_n(t) \frac{\binom{n+1}{2}}{2N}\exp{\left(-\frac{t\binom{n+1}{2}}{2N}\right)}$$ ...
0
votes
0answers
14 views

Average Gain, Trading Correlation

The following journal article seems to suggest Over a $5$-minute period there is a correlation between returns The average return is $0.037\%$ The average daily gain is $0.59\%$ Would anyone know ...
0
votes
1answer
19 views

Having trouble calculating expected value?

In a mall, a survey found that the number of people who pass by JCPenney between 4:00 and 5:00 pm is a Poisson random variable with parameter λ = 100. Assume that each person may enter the store, ...
0
votes
1answer
29 views

Need help finding joint probability density function

Let X and Z be independent random variables with X uniformly distributed on (−1, 1) and Z uniformly distributed on (0, 0.1). Let $Y = X^2 + Z$. Then X and Y are dependent. Find the joint pdf of X ...
0
votes
2answers
4k views

Variance and Standard Deviation of multiple dice rolls

I'm trying to determine what the variance of rolling $5$ pairs of two dice are when the sums of all $5$ pairs are added up (i.e. ranging from $10$ to $60$). My first question is, when I calculate the ...
0
votes
1answer
32 views

Conf. Interval for a simple mixture of Normals

Imagine there's a prob. p=0.5 of choosing one machine or the other to take some measurements $X$ for an experiment. One machine ($N(\theta,10)$) is much less precise than the other($N(\theta,0.1)$) ...
0
votes
1answer
21 views

Is $2\bar x(1 - \bar x) - \sum_{i=1}^n 2 x_i (1-x_i) = 2 \sum_{i=1}^n (x_i - \bar x)^2$ true?

In Nei 1973, right after equation (9), the author says: [..] it can be shown that $H_T = 2\bar x(1 - \bar x)$ and $D_{st} = 2 \sigma_x^2$, where $\bar x$ and $\sigma_x^2$ are the mean and variance ...
0
votes
2answers
645 views

Find the Posterior distribution- prior: $\exp(1)$, likelihood: $poisson(\lambda)($

I have a prior $\lambda \sim \exp(1)$ and a likelihood $X \sim poisson(\lambda)$, and I observed in a sample of $n=5$ a mean of $3$. What is the posterior distribution of $\lambda$? Here is my ...
0
votes
2answers
21 views

Does this hold in every case, and if only this one, why? Expectation, mean of random variable.

Characteristic function of random variable $X$ let us denote as $f_X(t)$ and $EX$ it's mean or expectation. Does the following hold in all cases, because it keeps coming up and I don't know why it is ...
1
vote
1answer
20 views

clarity on empirical probability

My definition of empirical probability: Let $(X_1,\ldots,X_n)$ be a random sample of i.i.d random variables. Then, the empirical probability for a subset A contained in the sample space $\Omega$ is $...
1
vote
0answers
18 views

Find conjugate prior of an exponential family distribution

I read on Wikipedia that all exponential family distributions have conjugate priors. I have not, however, been able to find a reference that describes how to find it. So given $$f_X(x\mid\theta) = h(x)...
1
vote
2answers
46 views

How was statistics formulated [on hold]

I'm sorry for the naive question, but I've only been taught statistics at a high school level (extremely basic) and it always remained a mystery as to how certain concepts in statistics existed (...
0
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0answers
7 views

How to use unsupervised classification to “test” labels of supervised classification

If i know $k$ labels i can do a supervised classification. Have any sense perform also unsupervised classification and made a prediction table to understand if the new $k$ label of unsupervised are ...
1
vote
0answers
12 views

Finding “the” Marchenko-Pastur distribution in the original article of 1967

I am looking at distribution properties of eigenvalues of sample covariance matrices. Following the Wikipedia article on the Marchenko-Pastur distribution: Let $X$ denote a $M \times N$ random ...
2
votes
1answer
19 views

OLS under Mean Independence

Assume mean independence of the error term. Show that the OLS estimator is unbiased. Hint: use the Law of Iterated Expectations I am not quite sure if I understood the concept of 'mean independence'....
0
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0answers
16 views

Combinations over tree nodes

Assuming to have a generic tree, how can I calculate all the possible combinations of 1,2,3...n nodes (with n that represents the number of nodes at a certain level of the tree) that can be generated ...
0
votes
0answers
31 views

Computing the mean and variance of the ratio of two Laplace variables

I know that Laplacian distribution function is defined as follow $$ f(x)=\frac{b}{2}\exp(-b|x-\mu|) $$ Also, I know that the mean and variance for the ratio between two normal variables like $$c=\...
0
votes
0answers
16 views

What is the statistical distribution that models people waiting in a queue e.g. at a bank. [on hold]

I am looking for the statistical distribution that most closely models the real world scenario of people waiting in a queue e.g. at a bank. Please also indicate what assumptions are made about the ...
-4
votes
0answers
22 views

Given truck A arrives at a random time between 9am and 11am, and truck B arrives at a random time between 10am and 12pm (noon). [on hold]

I tried to solve the problem using hints in a previously posted here at What are the odds that truck a arrives before truck b?. I would like a detailed solution.
1
vote
2answers
29 views

Standard Error - Statistics [on hold]

A box contains twelve tickets labeled with numbers. The number on the tickets are: -10,-6,-4,-3,-2,1,3,4,4,5,8,8 a) The standard error of the sample sum of the ticket labels in 5 independent random ...
3
votes
2answers
1k views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
0
votes
2answers
25 views

statistics- probability question [on hold]

Let E be the event that a corn crop has an infestation of ear worms, and let B be the event that a corn crop has an infestation of corn borers. Suppose that P(E) = 0.24, P(B) = 0.16, and P(E and B) =...
0
votes
1answer
45 views

normal distribution formula conventions

I sometimes see people write normal distribution formula like this, wondering if $G$ means Gaussian? And what does $C$ means here? Thanks. $G(\mu, \sigma)$ $\exp(\mu + C(\sigma))$ thanks in advance,...
4
votes
3answers
374 views

Does introducing penalties for getting true/false questions incorrect result in higher skill penetration (less luck/variance)?

A student is asked to answer 50 true/false questions and he would get 35 right and 15 incorrect if he had to put his best guesses for each question down. Now, for each question he has a certain ...
1
vote
2answers
33 views

How to compute variance of a conditional expectation and vice versa

I am trying to use the law of total variance which is $$\operatorname{Var}(X) = \text{Var}(E(X\mid Y)) + E(\operatorname{Var}(X\mid Y))$$ But I honestly have no idea how to compute either one of ...
0
votes
0answers
25 views

Concentrated Likelihood

I am working on Panel Data models and I am having some issues to obtain the concentrated log-likelihood function for the following model. $$ y_{it} = \delta'x_{it} + \beta_i'f_t + u_{it} , i = 1,...,N,...
0
votes
2answers
42 views

How to correctly count the probability for a computer game situation? [on hold]

Imagine we have the following situation in a computer game: One player has two minions with 30 and 6 hitpoints correspondingly. Another player casts a spell which does 12 times 1 damage (for each of ...
-1
votes
0answers
18 views

Log-Poisson distribution. Literature?

If I have a Poisson distributed random variable $X$, then adding two to get $Y=X+2$ I can define the new rv $Z=\log(Y)$. I would call this a log-Poisson distribution or better exp-Poisson ...
2
votes
1answer
604 views

Confidence interval of quotient of two random variables

I have random variables $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$, with $n$ a large integer. All pairs $(X_i, Y_i)$ are independent and identically distributed, but every $X_i$ and $Y_i$ ...
1
vote
2answers
18 views

Number of outcomes, if having known the distinct numbers and number of choices

This question came to me, when I was solving another relavent question in my class: We have $N$ distinct numbers, say $P(X=i)=1/N$, with $i=1,...,N$. We choose $n$ (known) numbers from them (with ...
1
vote
0answers
96 views

Joint probability with constraint [on hold]

Let's say that one is conducting an experiment with 8 units and 4 units have to be assigned to treatment. Assuming all units' respective treatment assignment probabilities are greater than 0 and less ...
0
votes
0answers
24 views

n-dimensional Archimedean copulas

I am studying Nelsen's book Introduction to copulas and I want to extend $\max(1-[(1-u)^{\theta}+(1-v)^{\theta}]^{1/\theta},0)$ to an $n$-dimensional copula. The problem is that it seems that the $\...
0
votes
2answers
42 views

How to calculate f(x, y) = rnd(x) > rnd(y)

I ran into an interesting answer on gamedev. He uses an interesting formula and ran it 100.000 time to get the average percentages. The formula is as following: For ...
0
votes
1answer
32 views

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$

Find $E(Y)$ and $Var(Y)$ of $\log Y \sim N (\mu,\sigma^2)$ I tried solving this in 2 different ways. The second way is what I am stuck on: 1st Way: Let $Y=e^X$ where $X \sim N (\mu,\sigma^2)$. ...