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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Convergence of probability

So I am getting ready for my first econometrics exam, and we have a lot of these plim (probability limits). Looking at the definition, I have that as n goes to infinity, ...
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probability questions (wait time)

Let T be the waiting time in hours when we order a manually scheduled phone call to Timbuktu. The variable T has the probability density (picewise function) f(t)=k(4-t) ...
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17 views

Simple T Test of difference of means

Suppose I have the following data: $$\bar x_1=15$$ $$\bar x_2=12.5$$ $$\sigma_1=1.8$$ $$\sigma_2=1.75$$ $$n_1=800$$ $$n_2=3000$$ And I want to test: $$H_0:\mu_1-\mu_2>0$$ $$H_a:\mu_1-\mu_2<0$$ ...
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The critical value table

how can I derive the number of critical value table?!Is there any specific formula ?I have searched allover the internet but I didn't mange to find any.
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Show Y is location-scale if $\sigma > 0$ is unknown

Let X be a random variable having the gamma distribution with shape parameter $\alpha$ and scale parameter $\gamma$, where $\alpha$ is known and $\gamma$ is unknown. Let $Y= \sigma $ log $X$. Show ...
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52 views

Joint probability density of x and y

if the joint probability density of X and Y is given by $f(x,y)=\frac{1}{y}$ for $0<x<y,0<y<1$ 0 else where find the probability that the sum of values X and Y will exceed 1/2. So this ...
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42 views

Find $P${$min_{i \neq j}|R_i-R_j| \geq d$}, where $R_1,…,R_n$ are uniform on line with length L

If n points $R_1,...,R_n$ are picked independently and with uniform density on a straight line of length L, find the probability that no two points will be less than distance d apart; that is, find ...
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Stats questions! Please help

The time an american worker spends on the computer per workday is normally distributed with the mean of 3.1 hours and standard deviation 0.5 hour. Find the percentage of workers who spend less than ...
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Mean and Variance of X with possible outcomes

I roll a six-sided die until I get a 6. Then I roll it some more until I get an even number. Let X be the total number of rolls. So here are some possible outcomes with the resulting value of X: 24 1 ...
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Probit regression model: Construction of weighted least squares algorithm

I'm posting a difficult general linear model question which I would like to solve. Question: Consider a probit regression model for $y \in ${$0,1$}:$E(y|x)=\Phi(x'b)$, where $\Phi$ is the standard ...
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How to show that correlation is equal to zero from probability density function?

Let X and Y be independent standard normal random variables, that is, they both have probability density function given by ((1)/(2pi)^.5)*(e^-((t)^2/2))dt Let U = X + Y and V = X − Y . Show that ...
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29 views

Injective functions and sufficient statistics

I'm trying to prove that for a random sample $X_1,\ldots,X_n$ that depends on $\theta$, if $T$ is a sufficient statistic for $\theta$, then so is $T'=f\circ T$, for any injective function $f$. My ...
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What is statistical fault localization?

I heard a talk, and was wondering what it is. There seem to be articles that point to the methods of this, but what is it? Either ELI5 or in depth answers are appreciated.
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Error of standard deviation of a Bradley Terry model

I have a set of pairwise comparisons which I can fit the Bradley-Terry Model to. I am fitting it using a simple iterative algorithm according to Algorithm 1 of A Generalized Bradley-Terry Model: From ...
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2answers
44 views

Having some trouble understanding Probability Distributions

The question is : If X has the Distribution Function F(x) = { 0 for x< -1 , 1/4 for -1 <=x<1 , 1/2 for 1<= x <3 , 3/4 for 3<= x < 5 , 1 for x>=5 } find a) P(X<=3) ...
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Independent set of points in a square.

Suppose I select points uniformly at random in $[0,1]^{2}$ and two points share an edge if their euclidean distance is less than $r$. Suppose I have $n$ points $v_{1},v_{2},...,v_{n}$ selected in this ...
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Comparing MSE of two estimators

Let $\hat{\theta}_1$ be an estimator of $\theta$, which is known to lie in the interval $(a,b)$. Define an estimator of $\theta$, $\hat{\theta}_2$, by $\hat{\theta}_2=\hat{\theta}_1$ if ...
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14 views

Unweighted Mean of Percentages

I have come across an interesting problem, which I'm sure there must be a standard solution to, but lots of Googling hasn't yielded much (which may mean I'm well off track...). I am trying to ...
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45 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
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29 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
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Practical convolution of poisson and log-normal distribution

Hi guys, Im trying to make a Loss Distribution (in Excel), following the "Loss Distribution Approach". I do understand that the main idea is that we have a severity distribution and a frequency ...
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statistics problem, where did I mistake?

I searched interesting problem about statistic from http://www.mast.queensu.ca/~stat353/resources/pastfinals/final12sol.pdf $$ $$ But at the question No.2, I have some problem the red box $$ $$ ...
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What real life statistician's job look like?

I have recently finished statistics course and would like to know if statisticians really do what we covered in the course (usual college level stat course material). The course made me interested in ...
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31 views

How to Find optimal number

Say I am tracking my sleep for 1 week and these are the number of hours I sleep each night: (5, 6, 9, 4, 8, 9, 6) everyday that I track my sleep I am also taking a test that measures how well my ...
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probability density function question

The diameter of grains of sand from a sand pit, measured in mm, can be considered a continuous random variable X with probability density (picewise function) f(x)=4(x-x^3), 0<=x<=1, otherwise ...
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collection and analysis of data and the use of inferential statistical methods

Task 2: Global Warming The purpose of this task is to investigate statistically whether there is a significant change in temperature over time. Go to the same website: ...
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How do I use r code to solve for probability of normal distributions?

I don't understand which r code I am supposed to be using to figure these problems out. A brief explanation of what the code is doing would be amazing. The problems below are two different ...
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Generalization of the Glivenko-Cantelli Theorem

The classic Glivenko-Cantelli Theorem states that $$ \sup_{t}|F_{n}(t) - F(t)| \longrightarrow_{a.s.} 0 $$ where $F_{n}(t)$ is the empirical cdf. Looking at the proof of the theorem, it seems to me ...
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Soft question: Distribution of the kth powers of normal random variables.

If $X_1,..,X_n$ are standard normal random variables then it is knows that: $\underset{i=1}{\overset{n}{\sum}} X_i$ is a normal random vairable and $\underset{i=1}{\overset{n}{\sum}} X_i^2$ is a ...
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25 views

Finding the MLE of $\alpha$ from the following.

Given: $f(x) = e^{-(x-\alpha)}$ for $x \ge \alpha$ and $0$ for $x < \alpha$. Attempt: $L(\alpha) = \prod_{i=1}^n e^{\alpha - x_i}$ and so $l(\alpha) = \log[L(\alpha)] = \cdots = n \alpha - ...
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68 views

Calculating the expected winner of a Penney's Game using a Markov Chain.

I am trying to calculate the probability that one sequence of coin tosses is more likely to win than the other in a game of Penney's. The sequences are: HTHT and THTT. So far I've come up with the ...
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48 views

Calculate 90th percentile of F distribution using “rchisq” function in R programming language

I am working on a problem for my Data Analysis class and I have to figure out a way to calculate the 90th percentile of the $F(3,5)$ distribution using the "rchisq" function in R(programming ...
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Determine the MLE and method of moments estimator of $p$.

Given: $f(x) = p^{x-1} (1-p)$ for $x = 1, 2, 3, \ldots$ Attempt: For MLE, I successfully got $p = 1 - \bar{x}$ but have no idea how to calculate methods of moments. I already read various materials ...
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Present a combinatorial argument for the identiy $\sum^{n}_{k=1} k\binom{n}{k} = n\cdot 2^{n-1}$

This is a question in my textbook that does not provide a solution. Any help on a solution? Consider the following identity: $\sum^{n}_{k=1} k\binom{n}{k} = n\cdot 2^{n-1}$ Present a combinatorial ...
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Change of variable pdf inverse function

I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = ...
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maths probabilities questions

n will not exceed 50. a)Consider a random collection of n individuals in a room. Suppose $P_n$ symbolizes the probability to have at least two individuals with the same birthday(born in the sane ...
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Is auto-correlation a particular and simple form of pattern in time series?

In a time serie context and considering the sign of the variation of the variable auto-correlation means that for couple of measures at (time t, time t+1) the ...
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Is the sample variance for normally distributed data equivalent to the sample variance for data following a bernoulli or binomial distribution?

For normally distributed data, sample variance $s^2$ is: $$s^2 = \frac1{n-1}\sum (Y_i - \bar Y)^2$$ For a bernoulli distribution, sample variance is: $$\frac{\hat p \hat q}{n-1}$$ $\hat p = ...
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sampling distribution with mean equal zero and variance equal 1

$if X\sim N(0 , 1) $ with $ \mu = 0 $ and $\sigma^{2} = 1.$ show that $M_{X}(t) = \ell^{\mu t + \frac{\sigma^{2}t^{2}}{2}}$ i know that $M_{X}(t) = E(\ell^{xt})$ pls help
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Theory of hierarchy

In statistical physics we consider a system of a huge number of particles. Each particle on its own is characterized by parameters (like mass or charge) and dynamical quantities (like momentum, ...
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29 views

Probability: Independent events (exactly, at most)?

we have four aunts going to a party: Albertine (0.3) Karoline (0.5) Makronelle (0.8) Petronelle (0.9) a) find the probability that exactly two of the four aunts come to the party. b) find the ...
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Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
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Does it matter here that random variables are jointly normally distributed?

My lecture notes ask the following (true/false) question on understanding: Jointly normally distributed random variables are independent iff they are uncorrelated. I don't quite understand what ...
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Proof an edge in a geometric graph

Suppose i take two random uniformly distributed points $X_{1},X_{2}$ in $[0,1]^{2}$. In addition i connect $X_{1}$ and $X_{2}$ by an edge if $||X_{1},X_{2}||_{\infty} \leq r$ where $0<r<1$ and ...
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Showing that $-\ln{X} \sim \exp{\alpha}$ for $X \sim Beta(\alpha, 1)$

The CDF for $X \sim Beta(\alpha,1)$ is given by: $$F(x) = \frac{\int_{0}^{x}t^{\alpha-1}dt}{\int_{0}^{1} t^{\alpha-1}dt}$$ I am given to understand that $-\ln{X} \sim \exp(\alpha)$ if $\alpha > ...
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covariance matrix in bivariate distribution

I struggle to understand how exactly you get the covariance matrix in a bivariate normal distribution. The reason is probably that I have no idea how to obtain it at all. In the exercise I have I ...
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45 views

Powers of random variables always well-defined?

Given the random variable $X$, is $X^{0}$ a random variable? Can we take the expectation $E(X^{0})$? Is $X^{0}$ or its expectation defined or undefined under any conditions (say, on the sample space ...
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X and Y have same distribution. Is the distribution of X-Y symmetric to 0?

I met this question when I went through non parameter test. I know X-Y may not be 0. Like X follows standard normal. So does -X. But X-(-X)=2X. I just wonder is X-Y (the pdf, if exists) symmetric to ...
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68 views

Is there a result that the density function for $\chi^{2}$ must be related to the standard normal density?

Suppose we have a random variable $X$ with the property that $-X = X$ (in distribution) and $X^{2} = \chi^{2}(1)$? I want to be able to conclude that $X \sim N(0,1)$. The probability density function ...
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In Discriminant Analysis (SPSS), why there are F values of the canonical functions?

Recently, I read a paper in which the author uses Discriminant Analysis, here is his words: "The discriminant analysis produced three significant canonical functions (Function 1, F(18, 2130) = ...