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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calculate expectation of MLE

It's a question about whether $\hat{\theta _{MLE}}$ is an unbiased estimator of $\theta$. n independent pairs $(X_{1},Y_{1}), (X_{2},Y_{2}),....(X_{n},Y_{n}), n\geq 3$, where $Y_{i}=\theta ...
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1answer
26 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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2answers
55 views

Finding the Probability of a Sequence of Numbers in Materials Testing

There are $n$ numbers on a wheel. For this example, let's say $n = 20$. You are going to spin $x$ times. For this example, let's say $x = 7$. The chances of spinning the same number all 7 times is ...
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1answer
32 views

Permutation algorithm for simulating random variables $X,Y,Z \in [0,1]: X+Y+Z = 1$ and $X,Y,Z \sim U(0,1)$

Edit: Sorry, I tend to jump back and forth between math notation and computer science notation....often to the chagrin of my more rigorous colleagues (and Math.SE folks ;-) Also, I accidentally ...
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11 views

How many 4 letter words 'LOGARITHM' if (1) repetetion allowed (2) repetetion not allowed

What is the difference between repetetion allowed & not allowed? & what are the answers of (1) & (2)?
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1answer
31 views

Question about sum of chi-squared distribution

I want to prove that the sum of two independent chi-squared random variables is a chi-squared random variable. I am supposed to only use the fact that if $Q$ has a chi-squared distribution with ...
6
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3answers
135 views

Chances of someone being of a certain gender at websites

I have 2 of websites and I know the chances of a visitor being a female or male. Let's say I have 2 website where the chance of a new visitor being a female is 80%. If the visitor comes on website 1 ...
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0answers
9 views

Likelihood function, log-odds for one sample

$x$ is a observation from a binomial distribution with n trials and probability parameter $ \pi $, where: $ x \sim \sim b(n,\pi)$ Let $\gamma$ be log-odds so: $\gamma = log(\frac{\pi}{1-\pi})$ I ...
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1answer
25 views

computing expectation of two arm bandit

assume you have a two arm bandit with one arm having a fixed, known probability of payoff $p = 0.6$ and another having an unknown payoff $q$, which is drawn uniformly from $[0,1]$. Each game the ...
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1answer
21 views

Calculate size and power of a given PMF

Let $X$ be a random variable having probability mass function $f(x) = \begin{cases} \dfrac{2+4a_1+a_2}{6}, & \text{if $x=1$} \\ \dfrac{2-2a_1+a_2}{6}, & \text{if $x=2$} \\ ...
0
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1answer
37 views

Probability of team winning, if it is trailing by 5 ponts after the first quarter

The score difference between home team and away team resembles a normal distribution where the mean is 1.5pts per quarter and the variance is 6 per quarter. What is the probability of the home team ...
3
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0answers
44 views

when can I say there is a relationship between events?

I am by no means a math expert, but I am analyzing very large weather data data for a computer science course .I am taking away all the weather related issues out of the analyzing and just looking ...
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0answers
27 views

What is the formula for dividing by a number with a variance?

If the average finish time of the $2014$ New York marathon is $4.580$ hours $± \space 0.889722$ hours and the distance is $26.219$ miles, what is the standard deviation of the average finisher's ...
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0answers
25 views

Normal distribution function:determine probability of a given point in Java

My statistics since high school is gone An I am struggling to find out a way to determine the probability of a given point in a Normal distribution in java. I see that Colt ...
1
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1answer
47 views

Find joint likelihood function of observations $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_m$

Let $x_1,\ldots,x_n$ be observations from a normal distribution with mean $0$ and s.d $s_1$. Similarly let $y_1,\ldots,y_m$ be observations from a normal distribution with mean $0$ and s.d ...
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1answer
17 views

Comparing sample and population standard deviation

I want to compute the standard deviation of some data points that I obtain during four series of experiments. For the first three experiments that I have conducted, the number of data points that I ...
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0answers
47 views

Statistics; Hypothesis Question

A die is tossed 120 times with the following result; Number of turned up: 1 2 3 4 5 6 Total Frequency 30 25 18 10 22 15 120 Test the hypothesis that the die is ...
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0answers
14 views

Hypergeometric distribution Expectation and exchangability of variables

The expected value for hypergeometric distribution is calculated using linearity property of expectation. Th expectation is $n r_1/r$ for $n$ trials $r_1$ type 1(say red) balls out of total r balls. ...
0
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1answer
7 views

Estimating unknown weights of various parameters in an equation

I have an equation which has some unknown weights attached to various parameters. None of the weights are known. However, I have a history of data available with me which can be used to predict the ...
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2answers
28 views

Poisson Formula yielding number too large and can't seem to figure out why.

So the formula is $ P(X) = \frac{e^-U \cdot U^X }{X!}$. My average rate of success is $\frac 1 2 $. The number I'm testing is 4. The question wants me to work out the chance of there being more than ...
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10 views

Chi sqaured table for degrees of freedom 616?

In order to check heteroskedasticity, we use the White's test. I tried to follow this method below, however, could not find a table with df=2016 and 95,5% confidence. I don't understand how we get ...
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1answer
24 views

Show this converges in distribution to 0

Let $\{ X_n:n \geq 1 \}$ such that $$f_{X_n} = \begin{cases} (n-1)/2 &\mbox{if } -1/n <x<1/n \\ 1/n & \mbox{if } n<x<n+1 \end{cases}$$ Show that this converges to $0$ in ...
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17 views

Problem related to variance of first passage matrix of a absorbing Markov chain

Consider the below computations taken from Kemeny/Snell Finite Markov Chains. Here $N=(I-Q)^{-1}$ calculated from some absorbing MC. $N_2$ is the variance matrix of $N$ and $N_{sq}$ is taken by ...
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0answers
78 views

Probability of starting to drink

My question seems to be very trivial, but I'm still stuck with it. It's about probability calculations. Suppose you have a number of people that start to drink water at some time point (event), ...
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1answer
26 views

How to Calculate Population from A given Set Of Sampels

I have a sample set of data collected using a SRS of books with IDs from 1 to 100. {90,60,6,39,46,26,16} Using this data how can I estimate the max, in this senario I know the max id, but what if I ...
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0answers
14 views

problem with derivation of PCA as minimizer of MSE

I am trying to learn proof of the Principal Component Analysis(PCA) as the minimizer of mean square error between all orthonormal bases in M dimensional space. in part of proof we see that: Now we ...
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0answers
25 views

Determination of a Lower Confidence Bound

Two stochastic variables $X$ & $Y$are normally distributed with $X\sim \mathcal{N} (0,1)$ and $Y\sim \mathcal{N} (2,1)$. We declare that $W = X + Y$ and $W$ is normally distributed with $W\sim ...
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0answers
22 views

How to rate a statistical variable?

I have lots of data and I'm trying to attach a rating to each of the variables depending on how closely the correlate with any subgroups. For example, suppose I have a bunch of people and they are ...
2
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1answer
40 views

Prove convergence in distribution

I need help with the following problem. Let $X_{n1}, X_{n2}, . . . , X_{nn}$ be independent random variables, with the same distribution as follows. Let for k = 1, 2, . . . , n och n = 1, 2, . . . , ...
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43 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
0
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1answer
22 views

estimate coefficients of $y = \alpha x + \beta y + \gamma z + \epsilon$

I know how to find $m$ and $b$ for $y= mx +b$, which is : $m= \frac{\bar{x}\bar{y}- \bar{xy}}{(\bar{x})^2 - \bar{x^2}}$ and $b= \bar{y} - m\bar{x}$ How can we estimate $\alpha, \beta, \gamma$ and ...
0
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0answers
27 views

how can I find outliers in a 2 vector data set

I have a two data set $(X,Y)$ where $X$ represents the angles and $Y$ represent the signals. $X$ is always correct because I increment it by coding $x=x+1$. However, $Y$ could be sometimes wrong ...
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1answer
18 views

Checking the consistency and Bias of $\frac{\sum X_i +\sqrt{n}/2}{n+\sqrt{n}}$

Let $X_1,\ldots,X_n$ be i.i.d. $B(1,\theta)$ random variables, $0<\theta<1$. Then, as an estimator $\theta$, check if $T(X_1,\ldots,X_n)= \dfrac{\sum_{i=1}^n X_i +\sqrt{n}/2}{n+\sqrt{n}}$ is ...
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0answers
24 views

Comparing two collection of points

I have a function $f(v)$ that takes a vector $v$ of numbers and spits out a function $c(t)$ which defines a looped path in $\mathbb{R}^2$. I also have another looped path in $\mathbb{R}^2$ that is ...
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0answers
24 views

Comparing two vectors based on order and ranking?

What I want to do is compare the ordering of variables determined by the ranking of each variable. For example: Say, I have a rating system that is made up of 5 different ratings - Excellent Good ...
0
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1answer
36 views

Adding a constant to a $T$ distribution: does it preserve the sample variance and sample size?

Question is as stated: If $T_1$ follows a $T$ distribution with sample variance $s$ and sample size $n$ and $T_2 = T_1 + k$, does $T_2$ follow a $T$ distribution with mean $\mathbb{E}[T_1] + k$ ...
1
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1answer
28 views

Standard Normal Distribution Transformation Z=lnY

I'm not sure if my approach to this problem is correct and I need help I need to apply $Z=\ln{Y}$ to the following standard normal distribution and then find the distribution of $Y$ ...
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2answers
19 views

Distribution of the product *not* independent Bernoulli distributed RV

Let $X$ and $Y$ be two real valued stochastic variables , and assume that both are Bernoulli distributed $$ P(X=1)=1-P(X=0)=p_1, \qquad P(Y=1)=1-P(Y=0)=p_2 $$ for $p_1,p_2\in(0,1)$. Note that we do ...
0
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1answer
21 views

Random variables in estimators

As I am a beginner, I am confused with how random variables come into play in estimators i.e. For example, while estimating mean, the estimator used is where all Xi's are random variables. But in ...
1
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1answer
43 views

PDF of the r-th Order Statistic (essentially need help with a derivative or combinatorics)

So I get why it is that the CDF of the $r$th order statistic is $$ F_{X_{(r)}}(x) = \sum_{i=r}^{n} \binom{n}{i}[F_{X}(x)]^{i}[1-F_{X}(x)]^{n-i} $$ but I'm not seeing how to prove that the PDF is ...
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0answers
45 views

How can non-binary data be tested for distinguishability from random?

Given a large string of Base-N digits (in other words, not necessarily binary), is there software or an algorithm that can output whether the data is reasonably pseudorandom or reveals a pattern? In ...
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0answers
22 views

Deriving the Expectation Maximisation algorithm for a mixture of 2 Gaussians

I have been attempting this question on EM-Algorithm for a mixture of 2 Gaussian's for 8 straight hours. I have gotten to the point where I keep repeating myself and expecting a different result. I ...
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1answer
29 views
1
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1answer
45 views

Integration limits and probability density

So I've got the density function for the 2-dimensional random variable (X,Y): $$p(x,y) = \frac{4}{3}xe^{-x-y} $$ when $ 0 < y < x$. Otherwise, it's zero. I am now interested in the density of ...
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1answer
25 views

Chi-square distribution and uniform distribution

I have been stuck with this problem for a long time I hope you can help me. random variable $X$ has chi-square distribution with 2 degrees of freedom random variable $Y=e^{-X/2}$ prove that random ...
1
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1answer
48 views

Conditional likelihood of continuously-combounded returns

The simplest possible asset pricing model ist the geometric brownian motion for asset price. Here the price $S_t$ solve the familar $$dS_t = (\mu +0.5 \sigma^2)S_t \, dt + \sigma S_t \, ...
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15 views

PDF of three random variables simplification

It is well-known that the PDF of two RV of the form $Z=\frac{X Y}{X+Y}$ can be represented by $\min(X,Y)$. and the corresponding equivalent CDF is therefore can be obtained by ...
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1answer
28 views

How to confirm/disprove a hypotesis

I have a list of quadruples in the following form (name, smokes, mother-smokes, father-smokes) (Andrew, Y, N, Y) (Jessica, N, N, N) ... and I would need to ...
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1answer
35 views

Find sample size required for hypothesis to hold true

A coach has made a statement that his players have bigger lung capacity than the average of the population of the same age which is $3.4$. (Normal distribution) The measurements yield the following ...
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0answers
21 views

What makes the Gaussian kernel so magical for PCA, and in general? [migrated]

I was reading about kernel PCA (1 2 3) with Gaussian and polynomial kernels. How does the Gaussian kernel separate seemingly any sort of nonlinear data exceptionally well? Please give an intuitive ...