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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Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
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2answers
42 views

Prove that $X_1 + … + X_r \sim NB(r,p)$

Let $X_1,...,X_r$ be independent random variables with geometric distribution $X_i \sim Geometric(p)$. Then $$X_1 + ... + X_r \sim NB(r,p)$$ This is what I have tried: $$\begin{eqnarray} ...
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1answer
21 views

Maximum likelihood estimator for general multinomial

Let $(X_1,\ldots,X_r)\sim\text{multinomial}(n,(p_1,\ldots,p_r))$, where $p_r=1-p_1-\cdots-p_{r-1}$. The random likelihood is $Ap_1^{X_1}\ldots p_r^{X_r}$, for some non-zero $A$. The random ...
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Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why R^2 in the regression of Y on X = the sample correlation between X and Y the whole ...
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1answer
19 views

Let $Y_{1},Y_{2},…,Y_{n}$ be a normal distribution where $\mu =2$ and $\sigma = 4$. Find $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$

Let $Y_{1},Y_{2},...,Y_{n}$ be a random sample from a normal distribution where the mean is $2$ and the variance is $4$. How large must $n$ be in order that $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
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897 views

Statistics - Z-score confusion.

For z score, you are taking the sample value subtracting population mean and dividing it by std deviation. Is that correct so far? Now, the "sample value" is defined by an equation. In my scenario, I ...
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15 views

correlation coefficient is over 0.7

"correlation coefficient can be over 0.7 then we can say Two factors have some strong relation. Then What is the 'reason' that we can say like that? Explain it." I got that problem and I really dont ...
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1answer
27 views

Consistency of two measurements including means and standard deviations

This is a simplified version of a real life experiment where we have done two experiments attempt to measure the same quantity and we obtained the results $0.8 \pm 0.1$ and $1.2 \pm 0.2.$ (That's all ...
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2answers
29 views

Proof of the law of large numbers for higher moments

Let us work on some probability space $<\Omega,\mathscr{A},\mathbb{P}>$: I'm looking for (independent) proofs of two proofs, of the generalised weak and strong law of large numbers ...
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1answer
34 views

Lebesgue-Stieltjes Integral (Several Variables)

Let $\mathcal F$ be a convex set of probability measures or distribution functions and $F, G$ be two elements in $\mathcal F$. Let $T$ be a functional on $\mathcal F$ defined as follows. Note that $h$ ...
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47 views

Assistance with Bayesian Random Effects and Mixed Effects Models

I am looking to build either a random effects or mixed effects model for a project I am working on, but have had trouble finding good sources and understanding the general structure of the model. ...
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18 views

Distribution of the natural Statistic of an exponential family

I have to show, that the natural statistic of an exponential family also belongs to an exponential family. A distribution belongs to an exponential family, if its density has the following form: $ ...
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1answer
28 views

Conditional expectation, sum of exponential i.i.d variables. Integral of conditional density.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (exponetial i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. ...
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Poisson Binomial upper tail decreases when all succes probabilities decrease?

I have Poisson Binomial r.v. $K$ with success probabilities $p_{1},\ldots,p_{n}$, where $n$ is odd and $p_{i}\in(0,1)$ for $i=1,\ldots,n$. Let $F=\mathbb{P}[K\geq\frac{n+1}{2}]$, that is, $F$ is the ...
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1answer
19 views

Example of convergence in probability to a non-degenerate rv

Suppose the sequence of random variables, X$_n$, converges in probability to another random variable X. The condition requires that for any arbitrary distance, $\epsilon$, the probability that the ...
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11 views

MA(1) + White Noise = MA(1)

$ X_t $ follows an invertible MA(1) process: $ X_t= \alpha u_{t-1}+u_t $, where $ |\alpha|<1 $ and $ u_t $ is white noise with $ (0,\sigma^2) $. $ v_t $ is is another white noise process, ...
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14 views

Is this process deterministic?

Let be $$D_{t}:(\Omega, Q, P)\rightarrow (R,B)$$ $$w\rightarrow U(-1,1) $$ where U is an uniform variable. I have been told in class that $D_{t}$ is deterministic and lineal because if you fix $t$ ...
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6 views

Statistics formula for checking relativity of types

I have a table of percentages describing how different is an article, from articles of type A and articles of type B e.g. Let's say type A represents news reports from BBC, and type B represents new ...
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1answer
311 views

Correlation between complex random variables

I am struggling to find the correlation between two complex r.vs; X and 1/Y i.e. E{X*/Y}, where '*' denotes the conjugation operator. The complex r.s X and Y are correlated with each other with known ...
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2answers
458 views

How many white balls are there in the box most probably?

There are n balls in a box. Some of them are white. A ball drawn from the box turns out to be white. How many white balls are there in the box most probably? Alright, well i know there are originally ...
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2answers
299 views

statistics - expected and probable numbers

The task: The number of oil tankers, arriving at a certain refinery on one day, follow a Poisson distribution with the parameter $u = 2$. The present harbour facilities can serve 3 oil tankers a day. ...
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4answers
585 views

In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288

In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288 Please explain how to get this. I understand that we have GGGG => 4 ...
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Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
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1answer
19 views

If the tubes are shipped in boxes of $1000$, how many wrong-sized tubes per box can doctors expect to find?

The cross-section area of plastic tubing for use in pulmonary resuscitators in normally distributed with $\mu = 12.5mm^{2} $ and $\sigma = 0.2 mm^{2}$ . When the area is less than $12 mm^{2}$ or ...
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3answers
35 views

Four cards are drawn without replacement. What is the probability of drawing at least two kings?

Four cards are drawn without replacement. What is the probability of drawing at least two kings? is the below my answer correct or not?! since cards are drawn without replacement, ...
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1answer
20 views

confusion about equivalent covariance formulas

$Cov(X,Y)$ = $E[$((X-E(X))((Y-E(Y))]$ = $E(XY)-E(X)E(Y)$ = $E[XY-E(X)Y]$ = $E[(X-E(X))Y] So, $E[((X-E(X))(Y-E(Y))]$ = $E[((X-E(X))Y]$ . But intuitively, $Y$ and $Y-E(Y)$ are simply different. How ...
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26 views

Sum of bernoulli random variables

suppose Z is a random variable which is the sum of some random variables with bernoulli distribution: $Z=Z_1+Z_2+...+Z_m $ , $Z_i \in \{0,1\} ,$ $Pr(Z_i=1)=p=1-1/2^k$ or $1/2^k$ when k is an integer ...
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1answer
38 views

In a game, $0.38$ buy hotdogs, how large an order should she place if she wants to have no more that a 20% chance of demand exceeding supply?.

A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game with the Baltimore Orioles, the last before a long road trip. The ballpark's records from games played either ...
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354 views

Confidence interval for Uniform($\theta$, $\theta + a$)

I am encountering a difficulty with the following task. Have I made a mistake, or is this an inherent flaw in the notion of confidence intervals? (Other such flaws exist.) Consider a random sample ...
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2answers
68 views

Difference between Poisson and Binomial distributions.

If both the Poisson and Binomial distribution are discrete, then why do we need two different distributions?
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1answer
476 views

Comparing annualised volatility from monthly and annual data

I fear there is a very simple answer to this question and its killing me that I can't see it. I am interested in calculating historical volatility: I have monthly index values starting in Jan 2005 ...
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78 views

Show that $X_{(1)} \xrightarrow{P} 0$, where $X_{(1)}$ is the minimum of the random sample.

Let $X_1,X_2,\dots,X_n$ be a random variable sample from a distribution with pdf given by $$f(x)=\begin{cases} \frac{\alpha}{\theta^\alpha}x^{\alpha-1} & \,if \,0<x<\theta \\ 0 & ...
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2answers
59 views

Stroeker Problem: Sum of consecutive cubes being a perfect square

I encountered to following textbook problem in the book 'Introduction to probability' (p.34) by Blitzstein and Nwang. NO homework, but self-study ! Part a) is no problem, but b) struck me down. ...
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1answer
13 views

Binomial Probablity

A multiple choice test has 40 questions. Each question has five possible answers of which one is correct. Assuming that an individual has studied and has a 70% chance of answering a question ...
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1answer
580 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
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1answer
16 views

Trouble with Example for Convergence in Distribution

I am a bit confused by an example used to illustrate the concept of "convergence in distribution" Intuitively, this makes sense, since if we choose a large number of points from the distribution ...
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4answers
55 views

Negative binomial distribution - sum of two random variables

Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$ How do I go about proving this? I'm not sure where to begin, I'd be glad for ...
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1answer
10 views

Probabilities, with replacement

I have a question, and here it's: Three cards are drawn with replacement. What is the probability that at least one of the three cards is a king? i'm not sure if I can solve it without your help. ...
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6 views

Cox regression model: covariates and likelihood function

I can't understand the meaning of "covariate" in the cox model given by the function $h(t)=h_0(t)exp(\beta_1X_1+...+\beta_nX_n)$. I mean: are they variables or constants? I guess that in case they are ...
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Statistical match for given n-tuple in set of m n-tuples?

I'm having difficulty stating my question, mostly because I have very little statistics training and don't know where to start looking for my answer. Suppose I have $m$ $n$-tuples. I.e., let $S = ...
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1answer
526 views

Weighted Average of Percentage Increases

I believe I have quite a simple problem, but want clarification on whether it is the best method to use. Say I have have insurance line, and the net income (£) from this business for 2011, 2012, 2013 ...
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1answer
77 views

Number of lists at some Kendall-Tau distance

Given a ranked list (permutation) $R$ of $n$ elements, how many permutations of the same elements are there at Kendall-Tau distance $d$ from $R$ $(0 \le d \le \frac{n(n-1)}{2})$? Example: If $R = ...
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Hypothesis Testing- comparing z crit and score

I am learning some basic hypothesis testing for single population. Lets say: $H_0$: $\le$ 5 $H_a$: > 5 I have a Z crit of -1.65 and a calculated Z score of -.72 How would I determine to whether ...
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1answer
20 views

Evenly selected order statistic?

Recently, I'm studying Nonparametric methods, especially Order Statistics. And I saw a sentence which says that Order statistic has to be evenly selected. And now I'm wondering what 'evenly ...
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10 views

Independent random samples from normal populations produced the results shown in the table to the right

my question about part b I see that u_2 is more than u_1 but when I estimated t_0 which is 2.7252 , I found that t_0.1,7 < t_0 < t_0.05,7 equal 1.415 < 2.7252 < 1.895 it ...
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24 views

Probability with statistics using 'R'

I've been going through a few previous exam questions and came across this one and computed in the programing 'R'. ...
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53 views

Find the sample size necessary to determine the true proportion within 5% at the 10% level of significance

I do not understand what this question is asking and this question also states that no preliminary estimated proportion is given. Also find the preliminary estimated proportion of 70%.
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1answer
421 views

Predicting the number of orders from future customers

Tamara is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 450 customers ordered turkey, 375 customers ordered ham and 250 customers ...
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1answer
23 views

Binomial Distribution comparing percentages [closed]

If I know p of both N and n, then can I take the percentages from both upper and lower (2 standard deviations) thresholds of N to statistically say whether n is unusual? For example: N = 1948 and p ...
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Peak absolute variation of a Band-Limited Process around its current sample for a given horizon T

Is it straightforward to find a bound on the maximum possible absolute variation around the mean or the last sample of a band-limited process for a given time horizon like $T$? More specifically, how ...