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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How to compress data dimension

I have n points on the plane $(x_1, y_1) \ldots (x_n, y_n)$ - it's the points of one stroke. I want to reduce number of points without significant information loss. I read some information about PCA, ...
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20 views

Making sense of 1-dimensional data

I am collecting data of individual people. Each data is one dimensional, indicating a time difference of them doing certain things. Each person requires a different time value measured in miliseconds ...
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38 views

Working out percentages from other percentages - no raw data

I'm having some difficulty trying to figure out how to come up with a way to solve my problem. I have the following information: 12% of men do X activity 26% of women do X activity 28% of 25-34 years ...
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1k views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...
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4k views

Prove that Cov(X,Y)=Cov(X,E[Y|X])

I've been working on this problem for 3 hours now, and my complete lack of progress is getting disheartening. I've looked up definitions, proofs, and have even seen a solution for this particular ...
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1answer
218 views

PDF of $Y = \cos(\pi X)$

I've run into a practice problem where $Y = \cos(\pi X)$, and $X$ is uniform on $[0, 1]$, and I'm supposed to prove that the PDF of $Y$ is $1/(\pi\sqrt{1 - y^2})$. I know that for derived ...
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1answer
96 views

Confidence interval without standard deviation

From a list of stock mutual funds, 52 funds were selected at random. Of the funds chosen, it was found that 19 required a minimum initial investment under \$1,000. (a) ...
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2answers
500 views

Why is SSE called unexplained variation

Why is $$SSE=\sum(y-a-bx)^2$$ called the unexplained variation? I have real trouble understanding this concept which leads to the definition of the coefficient of determination. The books keep ...
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78 views

Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
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204 views

Least Square with homogeneous solution!

I've read somewhere that: $x=A^+b+(I-A^+A)Z$ is a solution for $Ax=b$ ,when is doesn't have a particular solution. where $A^+$ indicates the pseudo-inverse and $Z$ is an arbitrary vector!!! I know ...
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71 views

Calculating the “odds in”…

If I say to someone, "I'm thinking of a number between 1 and 100, try and guess it". Their first guess will have a 1 in 100 chance of being correct. If I say to someone, "I'm thinking of TWO numbers ...
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88 views

Unbiased estimator with conditional expectation.

Suppose that $X$ has a binomial distribution with parameter $N=1$ and $p=1/2$. Y, which is independent of $X$, has a normal distribution with mean $\mu$ and variance 1. Consider the estimator $\mu$ of ...
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1answer
92 views

Variance of unbiased estimator

Let $Y_1,Y_2,...,Y_N$ be a random sample from a distribution with probability density function $f_Y(y,\theta) = 2y/\theta^2$ if $0<y<\theta$ and $0$ otherwise. (a) Show that $W = 3\bar{Y}/2$ ...
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1answer
71 views

Why is it called the score of the log likelihood function?

Since the score of the log likelihood function is just the gradient of the log likelihood function, why give it a special name? Why not just call it the gradient?
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1answer
391 views

How to decompose a sum of Gaussian curves into its substituent addends?

I have a radar that can scan its surrounding. If there is a new object in a previously empty environment, it will cause a bell curve in the amplitude of the light waves coming back. If there are two ...
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71 views

Resources for Teaching High School Statistics

I am a student teacher looking for resources to teach high school Probability & Statistics (untracked). The second semester will be inferential statistics and will include these following topics: ...
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17 views

Determine similarity between atom distances in models

This is applied to bioinformatics, but I feel it is also a statistical question. I have the x,y,z coordinates of 392 atoms in a 392-atom molecule. There are 14 models of this molecule. In other ...
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92 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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190 views

Ranking algorthim for views/likes/dislikes

I have a table of data that records the number of views an item receives by users viewing the content and the number of likes and dislikes that the users gave: ...
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1answer
80 views

Statistical dependency tests

I have a lots of football data for 2010 and I'm interested in finding the dependency of the number of average goals per game on a) the competition (e.g. Premiership, World Cup, FA cup) and b) the ...
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3answers
115 views

Are there reasons to consider complex valued random variables? [closed]

In all examples I know, we always work with real valued random variables. Why don't we ever consider complex valued random variables? Can anyone give an example of such a thing? It seems that if ...
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1answer
304 views

Probability that the majority of 3 classifiers are wrong

3 classifiers. Each classifier has $0.7$ accuracy and makes its error independently. How do I calculate the probability that the majority of three classifiers are wrong? how I'm trying to solve it: ...
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20 views

Combining classes in a multivariate hypergeometric distribution

Suppose I have a population of size $N$ made up of people of different U.S. political parties: $K_R$ Republicans, $K_D$ Democrats, $K_L$ Libertarians, $K_G$ Greens, $K_S$ Socialists and $K_I$ people ...
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2answers
60 views

Are A and B conditionally independent

Are A and B conditionally independent given the class label? I calculated that $$P(A=1) = \frac{1}{2}$$ $$P(B=1) = \frac{2}{5}$$ $$P(A=1,B=1)=\frac{1}{5}$$ My answer is yes. I do it by anding ...
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43 views

GPS coordinates statistics with rotated X & Y axis

I have a set of GPS coordinates along with their bearing angle. These coordinates are within 15 m of each other and so I can use cartesian math to determine the center, std deviation of coordinates ...
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1answer
419 views

Mean and Variance of Multiple Choice Exam

A Multiple choice exam has 100 questions, each with 5 possible answers. One mark is awarded for a correct answer and $\frac{1}{4}$ mark is deducted for an incorrect answer. A particular student has ...
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1answer
230 views

Naive Bayes to Predict a class label

How do I use Naïve Bayes to predict a class label for a test sample $(A=1, B=1, C=1)$ I know Bayes Theorem is: $$P(C|A) = [P(A|C) P(C)]/P(A)$$ I have no idea how to do this, please help.
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1answer
64 views

Probability: Why assume an equal relevant magnitude of two mutually exclusive events for P(A|B)?

While reading Grinstead and Snell’s Introduction to Probability p.134, I came across the following: "Let Ω = {ω1,ω2,...,ωr} be the original sample space with distribution function m(ωj) assigned. ...
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1answer
128 views

Unbiased estimators for the moments of 2 not independent random variables

Let $X$ and $Y$ be two non independent random variables. Suppose to generate $n$ realizations of both variables, and indicate with $(X_i, Y_i)$ their values. Also, let's pose that $X$ and $Y$ are non ...
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0answers
67 views

Finding an unbiased estimator for function of Poisson

Let $X_1,...,X_n \sim Poi(\lambda)$ then unbiased estimator for $\lambda$ is obviously $\bar{X}$. What about $\tau(\lambda)=\sqrt{\lambda}$? Also how would one derive UMVUE for this lambda?
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2answers
134 views

Generating points from a standard Gaussian

I'm new to Gaussian distributions and I'm trying to generate say, $ N$ points from a $ M$ dimensional standard gaussian. What does this mean? How would I do this in matlab?
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1answer
358 views

Calculating the p value for a difference between bootstrapped parameters using 95% confidence intervals

I have bootstrapped model fits to some data, using the 2.5 and 97.5 percentiles to generate a 95% confidence interval. I know that if 95% confidence intervals do not overlap then there there is a ...
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35 views

Approximating binomial probability distruibution.

I am studying Statistical Methods in my PG course. I came across Binomial distribution and here a question is given somewhat like this: According to BSNL, 71% of internet users connect their ...
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187 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
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1answer
188 views

Good Probability Practice Problems

I'm looking for a good probability textbook with lots of worked out examples and problems to prepare for my course's final exam. I'm in an introductory probability class in college, and we've covered ...
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1answer
40 views

Constant Sample Variance

If the sample variance $S^2$ is constant, can we say the population is constant? If so, how to prove this?
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1answer
24 views

Predicting data in many dimensions

I have two matrices deriving from one matrix of the original data. One is the training, the other is the validation set. Each matrix has rows= examples, columns = featuers. The proportions are 65% vs ...
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1answer
30 views

Probability density problem

Suppose that on each day that I cycle to work, there is a probability 0.33 that I get wet because of rain. Suppose I cycle to work on 12 days, then what is the probability that I get wet in more than ...
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52 views

Random variable $Y$ has moment generating function $M_Y(t)=(0.7+0.3e^t)^4 $ let $W=2Y-4$ what is the moment generating function of $W$

Random variable $Y$ has moment generating function $$M_Y(t)=(0.7+0.3e^t)^4$$ Let $W=2Y-4$ what is the moment generating function of $W$? i don't understand how $W$ interact with the $M_Y(t)$ .
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1answer
88 views

Multivariate Probability Distribution Expected Value and MGF

Let $X_1, X_2$, and $X_3$ be random variables - continuous or discrete. The joint moment generating function of the random variables is given by \begin{align*} m(t_1,t_2,t_3) = E(e^{(t_1X_1 + t_2X_2 ...
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119 views

Significance Tests & Confidence Intervals for Non-normal data

I have 2 nonnormally distributed samples of different sizes (N1~=N2). To evaluate whether there is a significant difference between these samples, I used the Mann_whitney U test (ranksum in MATLAB) ...
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1answer
65 views

How to check linear independence

How can I check the linear independence of my variables? I have this system $Ax=b$ where $A$ is a $N \times 4$ matrix. I want to check the linear independence between the 4 variables in matrix $A$.
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1answer
947 views

How do I calculate the marginal probability density function of Y?

Let $\displaystyle{{\rm f}\left(x, y\right) = \left\{x\quad \mbox{if}\quad 0< y < {1 \over x}\right\}\ \mbox{and}\ 0}$ otherwise. I need to calculate the marginal pdf of $Y$. I know I need ...
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0answers
39 views

probability and statistics - reaching a consensus using a stochastic matrix

I am reading the article "Reaching a consensus" of Degroot: http://socsci2.ucsd.edu/~aronatas/project/academic/degroot%20consensus.pdf. I don't have background in the field. Most of what is said there ...
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1answer
109 views

The complexities of coin-flipping

A fair coin is flipped $M$ times; heads comes up $h$ times and tails, $M-h$ times. Question: If we flip the coin $N$ more times, what is the probability $P$ that at some point during those $N$ flips ...
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4answers
438 views

Why is the exptected value (mean) of a variable written using square brackets?

My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies ...
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2answers
522 views

Poisson Distribution Lambda, Probability, and Looking for Exactly k

Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with a rate of $ \lambda $ = 10 per hour. Suppose that with probability 0.5 an arriving vehicle will have no ...
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3answers
1k views

non-exponential family probability distributions and their uses

I was actually trying to find information on non-exponential family probability distributions. So many of the distributions that we study in statistics are members of an exponential family of ...
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1answer
81 views

Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean.

Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean. I have tried deriving the function , which i got $2e^2$ ...
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0answers
136 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...