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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Conditional Probability of Two Poisson RV's

Question: During a given year for a circus performer, let X represent the number of minor accidents, and let Y represent the number of major accidents. The joint distribution is: $f(x,y) = \Large ...
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14 views

ANOVA - A-Level statistics - Blocking factors

This is regarding an A Level statistics question. We are testing for a difference between four cleaning solutions (A, B, C and D). The test is run over 5 days and each day only four trials can be ...
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30 views

What is the expected value of cosine of a multivariate Gaussian?

Suppose $X \sim \mathcal{N}\left(\mu, \Sigma\right)$. How do I evaluate $\operatorname{E}\left[\cos \left(t^{T}X \right) \right] $ and $\operatorname{E}\left[\sin \left(t^{T}X\right) \right] $? Does ...
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Peak requests per second, The Social Network Movie's Face Mash

In the movie The Social Network, the Harward network crashes after facemash.com gets too much traffic. They mention 22000 page views during the 4 hours it was online. Viral websites usually follow an ...
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1answer
20 views

Finding the probability of destroying the target

Hi I am trying to do the following problem. Consider a guidance system which is designed to deliver a bomb to a target site, which we denote $(x_0,y_0)$. As with all things, the guidance system that ...
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50 views

How can I find the joint MGF of this density?

The joint density of the variables $X$ and $Y$ is $$f(x,y)={{e^{-y}}\over\sqrt{2\pi}}e^{-{1\over2}(x-y)^2}; 0<y<\infty,-\infty<x<\infty$$ a)Find the joint moment generating function of ...
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1answer
31 views

Conditional Probability with two subsets

Question: A man plans to ship six boxes. Two of the boxes are insured, while the other four aren't. Each package that is shipped has a 10% chance of being damaged. What is the probability that: ...
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1answer
21 views

Find the value for z(0.1) from a distribution table?

I'm doing a statistics course, and I thought I had no problems using distribution tables to find values. For example, for the Gauss distribution, if I want $\Phi(-2)$, I will do $1 - \Phi(2)$ because ...
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1answer
10 views

Odds of being the last pick in a draw, twice.

My brother's work did a draw where they had 27 participants. They would draw a name until 1 remained, who would be the winner. He did a mock draw yesterday and the real draw today and the same person ...
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1answer
21 views

What are strategies for maximizing functions--specifically for finding the MLE--without using the derivative (statistics)?

I know how to use the MLE, just I run into trouble when I can't maximize the likelihood function with its derivative. As an example, I know that the MLE for a uniform distribution with parameter ...
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1answer
28 views

Correlation of three values

I have a data set that considers three values, $x,y,z$. And I have three questions: What's the relationship between $x$ and $y$? What's the relationship between $z$ and $y$? What's the relationship ...
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30 views

Random Samples and Sample Variance Bound

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a population. Show that: $$\max_{1 \leq i \leq n}|X_{i}-\bar{X}|<\frac{(n-1)}{\sqrt{n}}S$$ Where we have the sample variance ...
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1answer
18 views

How to find the PDF for Y=|X-1|?

Given $X\sim N(0,1)$ and $Y=|X-1|$. Find the PDF of $Y$. I tried to discussed when $x>1$ and $x\le1$, but this gives me two different functions and I have no idea how to combine them. However, ...
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19 views

Geometry of Vector Random Variables and Joint Distribution

I'm not a statistician but have been trying to understand the following problem in my research: I have two $3\times 1$ random vectors $\mathbf{v}$ and $\mathbf{w}$, and a function ...
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1answer
46 views

Problems using Rejection Sampling method

I'm supposed to generate random numbers from the following distribution: $$ f(x) = \begin{cases} \frac{3}{4}(2x-x^2) &\mbox{if } x \in (0,2) \\ 0 & \mbox{else} \end{cases} $$ I'm given the ...
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0answers
82 views

Why is there no unbiased estimator for $\frac{1}{\theta}$ for Poisson Distribution?

Suppose that $X_1,\dots,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$. To do ...
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2answers
60 views

Find $P(Y> \max\{X,-X\})$

Given: $$f(x,y)=\frac{1}{π}; x^2+y^2<1$$ Find:$P(Y> \max\{X,-X\})$ I understood the the range is that of a circle but I have no clue where to begin. Kindly help.
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1answer
34 views

How to calculate the shortest interval, for $P ( X ≤ 1 . 645) = 0 . 95$?

The problem statement said: Based on the fact that $\Phi(1 . 645) = 0 . 95$ find an interval in which $X$ will fall with $95\%$ probability. Therefore: Since $P ( X ≤ 1 . 645) = 0 . 95, ( -∞ , ...
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25 views

Statistical Analysis of an AI Solving Raven's Progressive Matrices

I need to build an artificial intelligence capable of solving some simple Raven's Progressive Matrices. The gist of it is that you are given an analogy "A is to B as C is to blank" and numerous ...
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2answers
23 views

Is “non-random parameter estimation” the same thing as maximum likelihood estimation?

In one book and a few papers, mostly on navigational tracking, I have found reference to the method of "non-random parameter estimation" but this term is not on the Wikipedia and not in a lot of ...
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1answer
42 views

Degrees of Freedom in Covariance: Intuition?

If we say $Var(x)$ has $n-1$ degrees of freedom which are lost after we estimate $Var(x)$, this matches how $n-1$ observations are now constrained to be sufficiently close to the remaining observation ...
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1answer
24 views

how to tell how good the statistical data is

I am learning statistics for data analysis. I have a question regarding to compare two data sets of different standard deviation (STD). Assuming we are measuring some data on the same system with two ...
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20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
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47 views

The higher moments of truncated Gaussian

We assume that $$X \sim N(0,1/d),$$ where $d\rightarrow \infty$. For $\delta > 0$ sufficiently small. My question is, what is the correct order (in terms of $d$) of $$ ...
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15 views

Prove the generalized solution to the normal equations $X'Xb = X'y$

I am just learning how to write these statistical proofs with matrices. Please check and revise: Prove $(X'X)^-X'y$ is a solution to the normal equations. Let the solution to the normal equations be ...
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26 views

How to prove col spaces are equal $C(X)=C(Px)$

Here's a rough draft of this proof, can you look over and edit? Given $X$ is an $n$ x $p$ matrix. Prove $C(X)=C(Px)$. (I'm still learning mathjax thanks in advance for any typo / edits as well). ...
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22 views

Show the Statistic is Complete

Consider a random sample $Y_1,\ldots,Y_n$ of the Uniform Distribution on the Interval $[-\phi,\phi]$ I'm wondering how I can show that the Statistic $$ T(\mathbf{Y}) = ( Y_{(1)} , Y_{(n)}) $$ is a ...
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1answer
22 views

Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition. Oh my god! In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle ...
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14 views

SD probability problem involving chi-square distribution

Preparing for an exam in statistics, I have been pondering the following problem: Given that in country X 14 % of people hold a university degree, find the probability that for random sample of size ...
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1answer
22 views

Applying central limit theorem to moment generating function

A random variable X has moment generating function $e^{500t+500t^{2}}$ . Find P(27100 < $(X − 500)^2$ < 50200). I'm a little confused how exactly I can apply a moment generating function, to ...
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Average takeoff distances of a variety of aircraft

I'm honestly really confused about how to approach this problem. I notice there is a huge gap between the numbers considering one take off distance is as low as 400 and another is as large as 1495 ...
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1answer
30 views

Limit of Snedecor's F

Suppose we have a random variable $X$ such that $X\sim \dfrac{d}{n-d}F(d,n-d)$, with $d,n\in\mathbb{Z}$. What happens when $n\to\infty$? And when $d\to\infty$? I think when $n\to\infty$ then it goes ...
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19 views

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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1answer
49 views
+100

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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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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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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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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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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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
37 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 ...