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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Verify the joint probability function

I had a question I was hoping for some help on: There are 8 similar chips in a bowl: 3 marked (0;0), 2 marked (1;0), 2 marked (0;1), and 1 marked (1;1). A player selects a chip at random and is given ...
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Proving the sample variance has a chi squared critical value

Let $X_1, . . . , X_n$ be independent normal observations with means $µ = 0$ and variances $σ^2$. For testing the null hypothesis $H_0 : σ^2 = 1$ versus the alternative $H_a : σ^2 > 1$ show that ...
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1answer
9 views

Maximum Likelihood Estimator of scaled beta

I want to find the MLE of $X = \theta{Y}$, where $\theta > 0$ and $Y \sim \mathrm{Beta}(2,1)$. The density for $X$ is given by $$f_{\theta}(x) = \frac{2x}{\theta^{2}}$$ on $[0,\theta]$. It has ...
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1answer
7 views

Could you find initial distribution of markov chain by looking at transition matrix?

Can you find out Probability of Initial distribution of markov chain? For example If you are given transition matrix, Can you determine P(x0=0) P(x0=1) and P(X0=2) where state space is {0.1.2} ??
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1answer
13 views

Geometric Setting?

Is this a geometric setting: Record the number of times a player makes both shots in a one-and-one foul-shooting situation. (In this situation, you get to attempt a second shot only if you make your ...
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13 views

Statistics Marble Simulation

Given the jar containing 20 red, 5 white, and 10 blue marbles, Joey thinks a more interesting problem would be to find the number of marbles you would have to draw, without replacing them in the jar, ...
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9 views

Is AR(p) strictly stationar?

Good evening. Is it true that model AR(p) is a strictly stacionar random sequence? Model AR(p) is given by $X_{t} = \varphi_{1} X_{t-1} + \ldots + \varphi_{p} X_{t-p} + Y_{t}$ where $\{Y_{t}\}_{t \in ...
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7 views

bayes theorem and Conditional probability (Honest & Corrupt) [on hold]

The government of a certain country has installed a new device that monitors the activity of public officials working for an agency prone to corruption. There are only two types of public officials: ...
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6 views

Simulation model, reality check on sample data

I wrote a simple simulation in C (a next-event simulation) and I'd like an opinion on the consistence of the sample data. The model is that of a two nodes service, the user enter on the first one and ...
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22 views

Mean for seat allocation

There are a set of kids (let's say 30) asked to sit in a row of seats, leaving at least one empty seat between them until all seats are filled. At the end, how do I calculate mean of the fraction of ...
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1answer
19 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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consistency of OLS on misspecified AR(1) process

Suppose the true relationship in data is driven by AR(1) process as follows: $$X_t=\rho X_{t-1}+\epsilon_t\hbox{ , }|\rho|<1$$ and $\epsilon$ is a white noise of $(0,1)$ expectation and variance. ...
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1answer
26 views

Finding conditional expectation from system of equations

I have three equations: $$X_m = \beta_0 + \beta_1 \cdot X_I + \varepsilon_{BL}$$ $$W_M = X_M + \varepsilon_{MBL}$$ $$W_I = \gamma_0 + \gamma_1 \cdot X_I + \varepsilon_{RDI}$$ The $\varepsilon$'s ...
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9 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
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7 views

Simple linear regression; Beta(0) = 0

In simple linear regression, $Y_{i}=\beta_{1}+\beta _{1}X_{i}+\epsilon_{i}$, what are the implications if $\beta_{0}=0$ such that $Y_{i}=\beta _{1}X_{i}+\epsilon_{i}$, assuming that $X=0$ is within ...
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2answers
45 views

Random sums of iid Uniform random variables

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...
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1answer
17 views

Help finding a probability density function

I am having a bit of trouble with this: Let $U=Z^2$ where Z is the probability density function of the standard normal distribution. So, $f_z(z) = \frac{1}{2\pi} e^{\frac{-z^2}{2}}$ I want to use ...
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22 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
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2answers
74 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
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1answer
17 views

Covariance of $2$ variables

I am given two random variables $X$ and $Y$. I am also given that $\mathbb{E}(Y|X)=\mathbb{E}(Y)=\mu_y$ and $\mathbb{E}(X)=\mu_x$. So if I need to calculate the covariance of $X$ and $Y$, ...
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18 views

Law of Iterated Expectation Proof

I have a proof that needs to be done: $$ \mathbb{E}(XY) = \mathbb{E}[\mathbb{E}(Y|X)\,X] $$ So I start with the following \begin{align} \mathbb{E}(XY) &= \mathbb{E}(X)\cdot\mathbb{E}(Y)\\ ...
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15 views

Confusing on some concepts of sufficient principle

Im reading Chapter6 of Casella Berger's statistical inference that talks about sufficiency principle. I've been confused a lot by the definition of sufficient statistics, here it is: Basically, ...
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1answer
24 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
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1answer
20 views

Problem solving the standard deviation for a stochastic variable

Information: In a laboratory we have a vial of water that's being kept on 50 degrees Celsius. This is measured with the best thermometer in the world. The standard deviation on this thermometer is ...
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how are $t$ tables created? function relating $p$-value with $t$

I'm playing around with simulating data and comparing it to known data, and interested in how many $n$s I need in the simulated data to get significance and was wondering if there is an equation ...
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1answer
12 views

Finding the covariance of the sample variance and sample mean of a random sample with a standard normal distribution [on hold]

I am trying to work out the proof for $\operatorname{Cov}(s^2,\bar{x})$, of a random sample with $N(\mu, \sigma^2)$. I just don't know how to attack this problem and would really love some help.
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1answer
28 views

Probability and Statistics (Normal Distribution)

Having trouble with the last part of this question. Not sure how the man would divide his pile of vouchers? It seems that you could interpret this question in a lot of ways. Any tips would be ...
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11 views

Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
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0answers
18 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
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Categorical Survey Heterogeneity Measure?

A survey asks the user to assign ten stick figures into four categories: strongly oppose, weakly oppose, weakly support, strongly support. The survey has been completed, so it is too late to redesign ...
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1answer
26 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
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5answers
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Mutually exclusive AND independent event (help with examples) [on hold]

2 events cannot be mutually exclusive AND independent UNLESS one of the events are impossible. One example: The flipping of a coin details an independent event (flip#1 does not affect flip#2) ...
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1answer
27 views

Find the covariance of $Y_1$ and $Y_2$

I had a statistics question I was hoping for help on: Let $Y_1$ and $Y_2$ be discrete random variables with join probability function: $$f(x,y) = \begin{cases} \dfrac{y_1 + 2y_2}{18} & \text{if ...
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2answers
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Can we predict next sample using the existing samples?

Suppose that I have 18 data points and I'm sampling 3 data points each time. Suppose that I have 60 samples (each has 3 data points). Can we predict the next sample (of 3 points) from the existing ...
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23 views

Central Limit Theorem proof.

I am trying to understand the proof of the Central Limit Theorem in my book. However, I don't really understand what is going on. I know the proof is assuming that the moment generating functions of ...
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22 views

Problem involving statistics [on hold]

$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt\pi}2$$ $$\explain$$
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7 views

Sufficient condition for not having infinitely small modes in a distribution

I was reading the paper Optimal Throughput and Delay in Delay-tolerant Networks with Ballistic Mobility (http://dl.acm.org/citation.cfm?id=2500432), and found the following proposition (page 305): ...
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1answer
21 views

A starting lineup consists of 2 forwards, 2 guards and 1 center. How many different starting lineups..

A certain school has $4$ forwards, $4$ guards, $3$ centers and $1$ person who can play as either a forward or a guard. How many different starting lineups can be made? I came up with 2 answers to ...
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2answers
18 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
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1answer
14 views

geometric distrubtion and binomial confusion

I'm confused with this question - here I'm assuming our discrete random variable is $X$ which is "the number of washing machine breakdowns in a year" is this correct? If so, what is $p = 0.8$ ...
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1answer
19 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
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0answers
16 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
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Determining the actual number of observations in a dataset

I have two datasets one is a dataset with doctors in which I have the procedures they have performed at a given hospital where the actual number of procedures is not captured by this data since it is ...
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Proof of Optimality for Approximation of Probability Spaces by PCA

I have come across a theorem that states, that the $d$-dimensional subspace found by PCA is the optimal approximation of a probability space with such a plane, in the sense that it minimises the ...
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Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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16answers
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Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
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Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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2answers
34 views

Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
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1answer
27 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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2answers
24 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...