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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Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
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46 views

Joint p.d.f stats help [closed]

$X$ and $Y$ are random variables that have a joint p.d.f., given by $p(x,y)=cx^9y^6$ when $0\le x,y\le 1$ and $p(x,y)=0$ for all other $x,y$. Here $c\ge0$ is a constant, which you should find. What ...
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30 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = \frac{x_{i}x_{j}}{\sum_{i=1}^{n}x_{i}^{...
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315 views

Example of dependent but conditional independent

There are a lot of events that are independent and conditional independent. Is there any events that are dependent but conditionally independent?
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69 views

Joint pdf random variables

$X$ and $Y$ are random variables that have a joint p.d.f. given by $p(x,y)=2⋅\frac{(x+2y)}{3}$ when $0≤x,y≤1$ and $p(x,y)=0$ for all other $x,y$. Find the probability that $X<(1/3)+Y$. I'm ...
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36 views

Probability of Multiple Random Variables

Let $X_1$,$X_2$,$X_3$,$X_4$ be independent standard normal random variables and $Y=X_1^2+X_2^2+X_3^2+X_4^2$. Find the probability that $Y\leq 3$. I thought that you would be using some kind of ...
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25 views

Why is $1 - \operatorname{P}(\text{Type I error}) =\overline{\operatorname{P}(\text{Type II error})}$?

I understand that in hypothesis testing, an increase in $\operatorname{P}(\text{Type I error})$ will lead to a decrease in $\operatorname{P}(\text{Type II error})$. However, why does an addition of ...
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60 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any non-...
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62 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt \...
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137 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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221 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 \\x=2&0.1&0.25&0....
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143 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
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60 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
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91 views

how to find standard deviation when given a percentage?

I'm stuck on this question, can anyone help? An electronics company's biggest seller is a talking toy. Of the toys produced, 4% have a defective voicebox. A quality control tech randomly collects 250 ...
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49 views

marginal distributions of a joint pdf

find the marginal distributions of $$ f(x,y) = 2xe^{-y}, \quad 0 < x,\quad x^2 < y $$ I have drawn the support, with $y = x^2$. Not sure how to proceed (tried it on wolfram advanced calculus ...
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30 views

finding the cdf of the max of a joint distribuion [closed]

Random variables X and Y have a distribution given by the following pdf: fx,y (x,y) = 1, 0 < x < 1 , 0 < y < 2x ; and o otherwise find the cdf of Z = max(x,y)
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779 views

Find the probability that the difference between the sample mean and the true population mean wll not exceed 0.5 inch

an anthropologist wishes to estimate the average height of men for a certain race of people. if the population standard deviation is assumed to be 2.5 inches and if she randomly samples 100 mean, find ...
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38 views

Decomposition of sum of two independent random variables

Let $X$ and $Y$ be two indpenent r.v. How can I get an expression for: $$E[X|X+Y=a]$$ where $a$ is a constant? In other words, is there a general rule to recover the expected value of $X$ when all I ...
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29 views

Order statistics that are iid random variables

Given that $X_1$ and $X_2$ are independent identically distributed random variables with mean $1$ (exponentially distributed), find $\text{Prob}(X_2 > 2X_1)$? I'm not sure how to attack this, if ...
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49 views

Find Normalizing constant

let $f(x,\theta)=C_\theta \exp(-\sqrt{x}/\theta)$ where $x$ and $\theta$ are both positive. Find the normalising constant $C_\theta$. I get $C_\theta=\sqrt{2}/\theta$ but my book says $C_\theta=1/2\...
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1k views

Distribution of Chi-Square divided by its degrees of freedom?

I have the following: $$\frac{2n}{\chi^2_{\nu=2n}}$$ Does this simplify to be a $\chi^2_{\nu=1}$ distribution by any chance? Or is there a rule to get rid of the $2n$? Any help would be ...
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67 views

Establishing fairness in test grading

Consider a group of 75 students who sit an exam consisting of 20 open questions, and are then randomly divided into 3 groups of 25 students {A, B, C} for grading by 3 different persons. Let us ...
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Revenue significance test

I would like to ak if is it possible to make a significance test for revenue from A/B testing when I have these data: ...
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131 views

iid random variables (vectors)

If $(X_{1},Y_{1}), (X_{2}, Y_{2}),...,(X_{n}, Y_{n})$ denote a sequence of iid random variables from $(X,Y)$, can I say that each $X_{i}$ is independent from each $Y_{i}$? Or is it just for the vector,...
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282 views

Proving negative natural log of Beta($\alpha$, 1) distribution is an exponential distribution

I'm looking to prove that taking the negative natural logarithm of a Beta distribution with parameters $\alpha$ and $\beta=1$ is an exponential functions. I've found two different proofs, both of ...
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37 views

Use of the chi-square test with low expected value

I have a question about the use of the chi-square test. I've been told that use of the chi-squared test for independence requires that most of the expected values be over 5. What is worrying is ...
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1answer
104 views

What does “central value” mean?

How to calculate central value of the following sets: I'm thinking is the same as the median - is it? $\{-2, -1, 3, 5, 7 , 1, 3 , 6, 2 , -1, -5\}$ and $\{-2, -1, 3, 5, 7 , 1, 3 , 6, 2 , -1 \}$
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758 views

Reliability of linear regression to predict future

When we have a set of data, where X is the cause, and Y is the effect, we can use linear regression to predict values for Y, based on values of X. I have learned that you may only safely apply this ...
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96 views

Mean of data with uncertainties and confidence interval

My question sounds rather simple, but sadly I have not been able to find an answer online (which is rather strange as this seems very basic, so I apologize if I simply haven't looked well enough) The ...
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88 views

Finding error variance and confidence interval

Two new types of petrol, called premium and super, are introduced in the market, and their manufacturers claim that they give extra mileage. Following data were obtained on extra mileage which is ...
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45 views

Finding confidence interval

A random sample of $700$ units from a large consignment showed that $200$ were damaged.how can we find the $95 \%$ confidence interval for the proportion of damaged unit in the consignment.
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327 views

Calculating the variance, mean, and autocorrelation of a time series.

How can I calculate the mean, variance, and autocorrelation function: $$Y_t=5+Z_t+ 0.6Z_t-1$$
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44 views

question about probability problem

How is the last column calculated? I don't understand, and I don't understand the explanation. $P(A \cap B)$ is calculated by $P(A)P(B\mid A),$ right? How is $P(A\mid B)$ calculated? Thanks
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173 views

stdev and mean from gaussian fit vs. from classical formula

I have a set of data - measured speed of molecules in water. I made a histogram and fitted it with function $$A\exp\frac{(x-B)^2}{C}$$ calculating mean and standard deviation from values B and C If I ...
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89 views

Odds of winning more than 50% of many bet of different %

I made a bet with a friend and I would like to know if I'm ahead or not. We have a package of 6 games. Each game have a different probability of a team to win. If it's a tie 3-3, it's a push. If it's ...
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16 views

Integrate a PDF over a set

This is part of a proof of Chebyshev's inequality, but there's one line I'm just trying to clarify my intuition of: $f(x)$ is the pdf and $g(x)$ is a non-negative function. $$ \int\limits_{x: g(x) \...
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48 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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48 views

WHY?The Squared Euclidean Imbalance are equal to Fourier coefficients

I'm reading the classical paper about distinguishing attack, How Far Can We Go Beyond Linear Cryptanalysis ,Thomas Baign`eres, Pascal Junod, and Serge Vaudenay. The only proposition I don't ...
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48 views

Which hypothesis test to use

Two identical machines are used to make a special coin. We want to know if they have the same variability. A random sample is taken from each machine : $$ \begin{matrix} MachineA & 135 & ...
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48 views

Bound for deviation from mean of binomial random variable

Let $S_n$ denote a binomial $(n,p)$ random variable. The problem is to show that there is a constant $C$ depending only on $p$ such that for all $\epsilon > 0,$ $$P(|S_n/n - p| > \epsilon) \le \...
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2answers
33 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
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333 views

What to do with the boundary values of a bin in a histogram?

Suppose I want to make a simple frequency histogram of the following data: $$\{3, 3, 4, 5, 5, 6, 7, 7, 8, 10, 11\}$$ I'm supposed to use bins of size $5$, starting with zero. Here's my question: Is ...
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49 views

how to test a hypothesis when given population mean and standard deviation is known?

the mean weight of a 6 year old child is 49.3 lbs. another study indicated that in a sample of 196 six year olds the mean weight was 51.5 lbs. Assume a population standard deviation of 14. use the ...
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$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& \text{elsewhere}.\end{cases}...
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49 views

A circle with radius $r$ has $k$ points within its radius. What is the min number of points a square with side $2r$ need to maintain this?

Consider that I have a circumference $c$ with radius $r$ that has $k$ points within its distance. As shown by the following graphic: For perfomance issues, I need to normalize this circumference ...
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94 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta \...
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60 views

Mean & SD of Sampling Distribution

A population consists of $4$ numbers $\{0, 2, 4, 6\}$. Consider drawing a random sample of size $n = 2$ with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal ...
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42 views

probability question needs some help

suppose $X$ and $Y$ are independent and identically distributed random variables that are uniformly distributed on $[0,1]$ What is the PDF of $ W=Y-X $ i tried to draw a picture to illustrated it ...
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Deriving the variance of the sample mean $\mathrm{Var} (\bar{y})=\frac{1}{n}(1-\frac{n}{N})S^2$

For a population of size $N$ with a simple random sample size $n$ derive the formula $$\mathrm{Var}(\bar{y})=\frac{1}{n}\left(1-\frac{n}{N}\right)S^2$$ where $S^2$ is the population variance. Hint: ...
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76 views

Standard deviations or two variances [closed]

Two variances or standard deviations? Listed below are the numbers of years that popes and British monarchs lived after their election or coronation.Treat the values as simple random samples from a ...