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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Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence?
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Jensen's inequality in derivation of EM algorithm

I am going through the derivation of EM algorithm and got stuck on understanding the following steps: Notes showing EM algortithm derivation For the equality to hold, f(x) has to be an affine ...
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Concentration inequality for sum of squares of independent and identically distributed sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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17 views

How to Graph Relationship Between 3 variables

I have 3 variables that I'm interested in understanding the relationship between. ...
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Constructing a copula that satisfies the desired condition

Exercise 2.8 in Roger Nelson's An Introduction to Copulas asks the reader to construct a copula $C(u,v)$ not equal to $\max(u + v -1 , 0)$ that satisfies the property $$ C(u,u) = \max(2u - 1,0) $$ ...
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31 views

Probability that first and last are same gender?

There are twelve members on a team, $8$ women and $4$ men. Out of these $12$, a $4$-person relay team is being chosen. What is the probability that the first and last leg of the race are the same ...
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Sampling bias using addresses as frame

A selected address in the sample has no residents. Will this cause bias in the results of a survey where house addresses are used as the frame? I think it wouldn't, because an address with no ...
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1answer
14 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
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1answer
29 views

Probability of Type II error

I have these values: A sample of size 100 is taken from the population. Standard Deviation is 5 Mean 125 H0: $ \mu=$ 125 against Ha:$ \mu<125 $ Accept H0 if the sample mean ...
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17 views

Calculating 5 different ranges for people resource management

I am working on a project for my company. My team is building a project charter template. In this template needs to be a drop down that estimates how many full-time employee days(FTE) will be ...
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1answer
19 views

Is multiplication of a correlated random variable and a independent random variable, an independent random variable

I have a random variable that is a multiplication of two random variables as bellow: $$A_n=B_n\times C_n$$ $B_n$s are identically distributed with zero mean and are correlated for different $n$s and ...
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1answer
26 views

Probability Type I error

I have these values: A sample of size $100$ is taken from the population. Standard Deviation is $5$. Average is $125$. $H_0$: $\mu =125$ Against $H_a$: $\mu <125$ Accept $H_0$ if the sample ...
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drawing cards in the deck [on hold]

Suppose 3 cards are drawn from a shuffled 52 card deck. The face cards are the Jacks, Queens, and Kings. Let A = {all diamonds} and B = {All face cards} Are the events A and B independent? ...
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Error in Six Sigma Rolled Throughput Yield Calculation

I've been looking at running some calculations on process yields and landed on Rolled Throughput Yield and Six Sigma. I had been using the product of all process yields in the past, no idea Six ...
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41 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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1answer
15 views

finding UMVUE $P ( X_1> t)$ [on hold]

Suppose $ X_1,\ldots,X_n$ are i.i.d. random variables with density: $$f(x_i;\theta)=\theta x_i^{-2}$$ $$x_i>\theta$$$$\theta>0$$ The smallest order statistic $X_{(1)}$ is sufficient and ...
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1answer
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How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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1answer
31 views

Confused with the power set of an integer

I am going through The Maximum Degree of a Random Graph by RIORDAN et al. On the second page, the notation $\mathbb{P}(\mathcal{D})$ is used which I assume the power set of the set $\mathcal{D}$. ...
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Deriving sample size using Hoeffding's Inequality

I want to use Hoeffding's Inequality to determine the necessary sample size $n$ to construct a confidence interval of $\epsilon$ and $\alpha$. I've consulted the Wikipedia article and am confused as ...
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How do you derive the confidence interval of linear fit parameters?

I have read a few ways of deriving the expressions for the parameters of a linear fit (i.e. slope and intercept) for a given set of values $X$ and $Y$. However, I have not found a treatment of: The ...
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14 views

How to determine the correct $\epsilon$ value for comparison of real values of a custom data set?

When comparing two real values we usually define an $\epsilon$ value and say they are equal if $|a-b| < \epsilon$. I evaluate Pareto approximations using different performance metrics also known as ...
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1answer
37 views

How come this Poisson formula equals 1

In Poisson Random Variable: $$\sum_{x=1}^\infty \frac{e^{-\lambda}\lambda^{x-1}}{(x-1)!}=1$$ Why does this equal $1$? What property is this?
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60 views

Coupon Collectors Problem with Packets: Clarifying Wikipedia

The Coupon Collector's Problem (CCP) is very useful in many applications. However, the "default" CCP is relatively simple: suppose you have an urn containing $n$ pairwise different balls. Now you want ...
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SQL joins and analysis

Say we have a users table and an events table and what sort of analysis can be done? Also, what is some SQL statements to describe the analysis of these 2 tables?
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3answers
29 views

How does one find the density of the $k$th ordered statistic?

Let $X_1,\ldots,X_n$ be $n$ iid random variables. Suppose they are arranged in increasing order $$X_{(1)}\leq\cdots\leq X_{(n)}$$ The first ordered statistic is always the minimum of the sample ...
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Proving Weak Law of large numbers by Markov's inequality

Hi I am trying to solve the problem 5.13 of the book Statistical inference by George Casella and Roger L. Berger. The problem is Formulate and prove a version of the WLLN with a weaker assumption ...
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1answer
39 views

Probability of a random Permutation [on hold]

Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last ...
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1answer
32 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
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30 views

How do you interpret conditional probability when two events are switched?

Before I pose my question, I want to emphasize that I am not seeking a homework help or steps on how to derive the answer, for I already know the solution, and how to get it. What I am seeking is, how ...
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1answer
50 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
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1answer
26 views

Urn probability replacement problem

An urn contains $10$ red and $10$ white balls. They are taken out at random one at a time. Find the probability that the fourth white ball is the fourth, fifth, sixth or seventh ball drawn if the ...
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Derive the asymptotic distribution of $\frac{2}{n(n-1)}\sum\sum_{i<j}|X_{i}-X_{j}|$

Derive the asymptotic distribution of Gini's mean diference, which is defined as $\frac{2}{n(n-1)}\sum\sum_{i<j}|X_{i}-X_{j}|$. This is an exercise of Asyptotic Statistics by A.W. van der Vaart. I ...
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Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...
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Relation of OLS and GLS risks

Let $Y=Xb+e$, where $Y$, $X$ and $e$ are random (usual linear regression model). Does it hold with some high probability that $$(Y-Xb)^T(Y-Xb)< c (Y-Xb)^T X X^T (Y-Xb)$$
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mann's test for trend

To test the null hypothesis that a sample $X_{1},...,X_{n}$ is i.i.d. against the alternative hypothesis that the distributions of the $X_{i}$ are stochastically increasing in $i$. Mann suggested to ...
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How many degrees of freedom exist in an agglomerative hierarchical clustering?

The computational complexity of generating an agglomerative hierarchical clustering from n vectors is $O(n^2)$ (calculating the pairwise distance matrix) dendrogram example However, the total number ...
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1answer
23 views

calculating the standard error of the mean?

The mean of a random sample of size $n = 35$ is going to be used to estimate the mean of a finite population of $N = 400$. Given that the population standard deviation is thought to be 9.355, what is ...
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Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
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28 views

Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
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Proving that a statistics is not sufficient (uniform case).

I am posting a similar question - in the previous one I put a wrong distribution, which changed the whole question. Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that ...
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statistic /normal distribution [on hold]

The exercise requires to determine the expected profit for one bottle of gass if the price is $40$ dollars for a bottle of gas , the cost of gas is $20[1+(x-100)]$ dollars/ kg of gas also the ...
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maximum likelihood estimators of a shifted gamma distribution?

i had this question in my exam but didn't know how to solve this apart from constructing the likelihood function and differentiating .but got stuck in the middle of nowhere.please help . the answer ...
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Why doesn't the approximate mean work for matrix having negative values?

Let $X$ and $W$ are $N \times N$ matrix where $x_{i,j} $ and $w_{i,j}$ are positive numbers. $for ( i =1;i <= N; i++)$ $ \qquad Mean_{x,i} = mean(X(i,:)) $ $\qquad for (j=1; j<=N; j++)$ $ ...
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2answers
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expection of random variable when the index also follows som

i don't have any clue to this question but because here the index also foloows a certain distribution please help?
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51 views

How to put my knowledge of probability and statistics to practice

Background: I am a masters student in stochastic analysis. My course is very theoretical, which in general is fine by me, it is what I enjoy the most. From the more data-friendly subjects, I have (or ...
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Probability of picking yellow after red.

I have a bag with $8$ red apples, $4$ green apples, and $5$ yellow apples. I select two apples without replacement, what is the probability that the second apple is yellow if the first is red? ...
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Estimation of using method of maximum likelihood

PDF $f(x;\theta) = \frac{x}{\theta^2} \exp \left ( - \frac{x^2}{\theta^2} \right )$ obtain an estimator of $\theta$ using the maximum likelihood method i think the likelihood function would be ...
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1answer
31 views

Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...
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IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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Why is the mean of the minimum of $100$ exponentially distributed random variables equal to $\beta$ divided by $n$?

Here's a question about order statistics, I can't seem to understand. Suppose a battery lasts $1,000$ hours. If I have $100$ batteries, why is it that the mean that the first battery will go out ...