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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Interpretation of the normalized squared Frobenius norm of correlation matrix

Let ${\bf R}$ be a $n\times n$ correlation matrix. I was wondering what is the interpretation of the following norm: \begin{equation} \frac{1}{n}\|{\bf R}\|_F^2 \end{equation} I know that is equal ...
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1answer
54 views

Distribution of the maximum of covariant random variables

I am looking to determine the distribution of $\max(X_i,Y_i)$ where $X_i = |A_i|^2$ $Y_i = \frac{1}{2}|A_i - A_{i-1}|^2$ Here $A_i$ is a complex vector with normally distributed real and imaginary ...
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2answers
63 views

Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
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1answer
16 views

How do I organize a multivariate linear model into matrices and vectors?

I'm reading through an SUU lecture on multivariate regression, and I'm guessing some linear algebra knowledge is assumed. However, I have very little familiarity with linear algebra. They start out ...
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0answers
32 views

$P(A=0, B=0)$ is what given the following graph?

Graph and Probabilities Given this graph and respective probabilities, what would be the value for $P(A=0, B=0)$? I computed $P(A=0, B=0)=P(A=0)P(B=0)=0.24$ because A & B are independent of D. ...
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1answer
29 views

Having trouble interpreting this statistics problem?

This problem seems simple enough but I can't seem to find a satisfactory answer. If $P(\text{A}) = .3$ and $P(\text{A} \cap \text{B}) = .25$, what are the possible values of $P(\text{B})$? At first ...
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1answer
22 views

Estimating probability based on past stats

We've a statistics that counts the number of events calculated over the past 20 years by year. The number of events per year can be a few (1,2,3). The idea is to calculate a distribution function. ...
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1answer
27 views

A question related to random variable with distribution function

Let X be a random variable having the distribution function $$ f(x)= \begin{cases} 0, & \text{if $x<0$ } \\ \frac{x}{2}, & \text{if $0 \leq x<1$}\\ \frac{3}{5},& \text{if $1 \leq ...
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12 views

Why if $e$ is independent to $\hat y$, then $\hat\beta $ will be independent of $S^2$.

$\bullet$ $X$ is a matrix $n\times p$, $\bullet$ $y=X\beta +\varepsilon$ where $\varepsilon\sim\mathcal N(0,\sigma ^2I_{n\times n})$ $\bullet$ $\hat \beta=(X^TX)^{-1}X^Ty$ $\bullet$ $\hat ...
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1answer
28 views

Find E[XZ] if X,Z are not independent and we do not know their covariance

I am very unsure on how to answer the following question. X and Y are independent normally distributed random variables with the properties listed below. How do I find E[XY] to find the covariance? I ...
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22 views

Expectation Maximization - question about the 2 different formulations of E-Step

I saw the following 2 formulations for the E-Step: $Q(θ,θ_t )=E_{θ_t} ( log(p_θ (X,Z))│X=x)$ $Q(θ,θ_t )= E_{Z|X,θ_t } [log(Pr_θ(X=x,Z))]$ I can't understand why they are equal.
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1answer
27 views

Probability that sample mean of iid Bernoulli random variables is close to true mean

Context: We have $X_i=1$ with probability $p$, $X_i=0$ with probability $1-p$, but $p$ is unknown. Given that the sample mean $Y_N=\frac{1}{N}\sum_{k=1}^N X_k$ is equal to $q$ after N observations, ...
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1answer
30 views

Mean number of unique choices given n people choosing randomly from a set of N elements

My question is similar to the birthday problem, but I can't seem to find a simple solution. The question (in a general form) is that, given a set of $n$ people who each choose elements from a set of ...
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1answer
38 views

Probability density function rules [closed]

Probability distribution function (pdf): \begin{align*} &P(-\infty<X<+\infty)=1\\ &P(x_0\leq X\leq x_1)\geq 0\\ &p(x_0)=-9.0? \end{align*} Source. Why does the third one make ...
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0answers
43 views

According to Chebyshev's rule, how many observations should lie within one and a half standard deviations of the mean?

Using the formula : $p = 1 - k^{-2}$ I calculated that $p = 1 - 1.5^{-2} = 0.56$ , which equals to $56\%$. Because I have $24$ data points I go ahead and solve the number of points is $56\%$ of ...
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2answers
27 views

If $P(ABC)=0.2$, are $A$ and $C$ mutually exclusive?

I was finishing up my statistics homework But I was unsure if I was thinking of the last problem correctly. It reads If $P(ABC)=0.2$, are $A$ and $C$ mutually exclusive? My thinking is that ...
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1answer
16 views

Finding chances with binomial

I have a huge problem that I can't solve, and it is keeping me up at night. I need to find the lower limit, where there is only 1% chance of getting a result above that limit. I have an example of a ...
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3answers
38 views

Different answers when I split a combination question into cases or solve in one step.

The students producing a school fashion show plan to have four pieces of music played. The music students have come up with 18 pieces: 6 for piano, 5 for recorder, and 7 for guitar. The students want ...
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1answer
29 views

Find the conditional distribution of several variables given their sum

The question is: Let $Y_1, Y_2, ..., Y_k$ be $k$ independent Poisson random variables with parameters $\lambda_1, \lambda_2, ..., \lambda_k$ respectively. Find the conditional distribution of ...
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1answer
37 views

If X is an independent variable… [closed]

If X is an independent variable, does it make X^2 an independent variable as well? I'd think that X^2 is not an independent variable because the occurance of one affects the occurance of the other. ...
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0answers
25 views

Can we illustrate pdf of Normal distribution with mean and variance following Normal by analytic expression?

I just wonder if pdf of Normal distribution with mean and variance which are normally distributed can be expressed in analytic formula, i.e. $ \mathcal{N}(\mu, \sigma^2) $ where $ \mu $ ~ $ ...
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0answers
34 views

Why is Method of Moments not the MLE for Beta Distribution?

This is a question from a statistics textbook: Suppose that $X_1, ..., X_n$ form a random sample from the beta distribution with parameters $\alpha$ and $\beta$. Let $\theta=(\alpha,\beta)$ be the ...
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1answer
38 views

How to calculate the integer $K$?

The number of policies that an agent sells has a Poisson distribution with modes at $2$ and $3$. $K$ is the smallest number such that the probability of selling more than $K$ policies is less ...
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1answer
23 views

calculation of median of grouped data

While calculating the median of grouped data of total frequency $N$, in order to find the median class which value should be taken into consideration to match against cumulative frequency : $\frac N2$ ...
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0answers
74 views

An algorithm for solving: $\max P(x_1, x_2, \ldots, x_n)$ in $O(m^2n)$ time

Assuming that $P$ satisfies: $P(x_i\mid x_{i−1}, \ldots, x_1)=P(x_i\mid x_{i−1})$ for all $i\ge2$. Given the probability $P(X_i=u\mid X_{i-1}=v)$ for each $u, v \in S$ (in an $m\times m$ table) and ...
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0answers
16 views

STATISTICS: What type of errors are included in the margin of error

Margin of error, as in the margin of error in the confidence interval. For example, I know non-response bias is NOT included in the margin of error, but random variation by chance is(?). What else ...
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0answers
17 views

Standard deviation and statistics

The concentration by volume of air mixing with methane is distributed with a mean of 20% and a standard deviation of 7%. I want to find the probability that the concentration is less than 30% or ...
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0answers
10 views

What is the best way of computing a POP (p-value of p-values)?

In evaluating PRNGs a "global" technique is running a test at k different seed and get k p-values which should be distributed uniformly. Even if the k p-values are good (e.g., within reasonable ...
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0answers
9 views

Related to statistics graph representation

The problem is related to statistics graph representation. The question is about representing 2 datas on graph. i am unable to understand how best the data can be represented. a) Two colleges A ...
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1answer
12 views

Statistic test for comparing two regression models

I'm having two linear regression models as follows: $y = a_1x_1 + a_2x_2 + c$ and $y = b_1x_1 + b_2x_3 + c$. I'm looking for a statistical test for proving which model is better. I've obtained the ...
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1answer
22 views

Determine the elements of a set, so that their mean, median and standard deviation are all equal to 3.

Determine the elements of a set, so that their mean, median and standard deviation are all equal to 3. The set has 3 elements, all elements are real numbers. I have already figured it out, that ...
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0answers
19 views

averages of distributions easy

Suppose I go on the beach looking for treasure, and that the amount of treasure I find on a given day is given by a poisson r.v. with mean 50. Also suppose on average 4/5 of the treasure I find is ...
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2answers
70 views

Escaping Prisoner Probability Question

Question: A prisoner is trapped in a cell containing three doors. The first door leads to a tunnel which returns him to his cell after two days travel. The second door leads to a tunnel that returns ...
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1answer
24 views

Understanding notation when finding the estimates in a linear regression model

I have been taught to calculate the estimates $\beta_0$ and $\beta_1$ (or $\vec{\beta}$) by using the following formula $$\sum_{i=1}^{n} y_i \vec{X_i}^T = \vec{\beta}^T\sum_{i=1}^{n} X_i ...
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1answer
27 views

Method of Moments Estimator for a non-standard distribution

How do you find the method moments operator for a non-standard distribution? The pdf is $f(x) = ax^{-(1+a)}$, for $x\geq 1$ and the proposed solution is $\frac{\overline{x}}{\overline{x} - 1}$, where ...
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57 views

How to identify what probability distribution this is?

in the context of a very long derivation (variational inference), by grouping the constant terms, I arrive at the following form that should be a probability distribution: $$ p(x) = C x^{-a} ...
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0answers
20 views

Hypothesis test given $\bar{x},s,\mu,\sigma$

I want to solve the following problem. A fuel is tested for cars in order to increase the number of km. A test take place in 16 cars and shows that $\bar{x} = 17.5\;\mathrm{km \cdot lt^{-1}}$ and ...
2
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0answers
26 views

expectation half-normal distribution or expectation Truncated Normal Distribution [duplicate]

I want to calculate integrals $$ \begin{split} \int_0^\infty x \exp\{ ax-b x^2\} dx &= \int_0^\infty x\exp\{-b(x^2-\frac{a}{b}x)\}dx\\ &= ...
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1answer
26 views

Pairing 10 people to meet each other each week

10 people would like to get to know each other better. If the 10 people meet once a week in pairs, how many weeks total to meet each person? If so how to determine which two people meet each week?
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1answer
10 views

Three Variable Transformation and Independence

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? ...
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0answers
13 views

Hypothesis testing with log likelihood ratio asymptotic behavior

I have an experiment whose result is a number in a finite set of m possible values. Let $\overrightarrow{X} = \left \{Xi, i = 1, 2, . . . , n \right \}$ be a sequence of results from the experiment, ...
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1answer
25 views

Compounding cash, at a gambling game with certain payout and your certain win rate.

Say there is a gambling game that pays $p=.70$ of bet per win and your win rate is $w = .70$. What is an expression for your account balance $P$, given a starting balance of $P_0$ if you're betting ...
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1answer
30 views

Estimating Number of Scratch Tickets Remaining

So, in my statistics class we discussed the recent huge jackpot, but we decided to focus on something more "trackable"—scratch tickets. A part of the assignment is to figure out how many tickets are ...
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11 views

Sufficient conditions for $|\hat{\theta_n}-\theta_0|=O_p(1/\sqrt{n})$

Let $X_1,\ldots,X_n$ be i.i.d. random variables defined on a probability space $(\Omega, \mathscr{F}, P)$ admitting some true unknown density $f_{\theta_0}$, with $\theta_0$ belonging to $\Theta ...
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1answer
13 views

unbiased estimator for sample covariance?

I'm new to statistics and and I need some help: Let $X_1,...X_n$~$N(\mu_x,\sigma^2)$, $Y_1,...Y_m$~$N(\mu_y,\sigma^2)$. All r.vs. are i.i.d and $\mu_x,\mu_y,\sigma$ are unknown I was told that ...
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0answers
46 views

statistics question (independent sampling)

You would like to know if pass rates are higher amongst individuals who take CFA preparation courses than amongst those who do not. You randomly sample 500 exam takers who took the CFA level I exam in ...
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0answers
32 views

Expected value of mean squares. ANOVA with one factor

currently i'm trying to find the expected value of the mean squares in an ANOVA. However, I think I have a mistake because the latter term is too long and confused me a little. Any help is welcome. ...
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21 views

Proof that the q-Gaussian is derived by maximizing Tsallis entropy

So I have to write a report paper for a course and I would like to prove that the q-Gaussian is the distribution which arises once one maximizes the Tsallis entropy. But I face difficulties in ...
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1answer
79 views

Scatter Plot with Marginal Dot Plot in R [closed]

I have found the following code in R that is supposed to solve the problem that I'm working on. The problem is to create a scatter plot with marginal dot plots. Here is the code: ...
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1answer
33 views