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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Delta method with uniformly convergent transformation

Let's say I have an estimator $T_n$ of $T$ and have established that $\sqrt{n}(T_n - T) = O_p(1)$ so that $\sqrt{n}(T_n - T)$ converges in distribution to some limit $G$. Let $f$ be differentiable ...
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151 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
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2answers
406 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
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61 views

Winning Percentages

A friend and I both play in an NFL pick league. His requires that he only pick 5 games per week. So far there have been 76 total games this year. His record is 18 for 25. My league requires me to ...
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1k views

Relationship between expected and actual value [closed]

Consider a zoo with two animals: an elephant and a lion. The elephant's expected lifespan is $70$ years, but it lives to be $65$ years ($SD = 7$). Likewise, the lion's expected lifespan is $22$ years, ...
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45 views

Curve Fitting and Multiple Experiments

Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the ...
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132 views

Using integrals to prove that the mean of the sampling distribution is the population mean

Let the random variables $X_1, X_2, \dots X_n$ denote a random sample from a population. The sample mean of these random variables is: $\overline{X}=\frac{1}{n}\sum\limits_{i=1}^{n}X_i$ I would like ...
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112 views

Find the marginal distribution of $(X,Y)$

Given the next robability density function$\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} \frac{1}{\pi^2(1+x^2+y^2+z^2)^2}\, dx dy dz $ ...
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562 views

Minimize and Maximize the variance of x

If we have 3 p(x=i) where p1+p2+p3=1. We also know that E(X)=2. How do i find the values of p1,p2,p3 that maximize the var(x) and also those that minimize it? Do we use the same method if ...
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115 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
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1answer
62 views

Conditional probability in continuous distribution

I'm struggling understanding how conditional probability for a continuously distributed random variable is to be calculated. The task is as follows: $f(t) = 1/8 * (4-t)$ for $0 < t <= 4 $ and ...
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41 views

Propogation of error in derived equation

I am trying to find the derive and find error in the derived equation from a set of experimental data. The equation is in the form $y = mx + c$. $\bar y_1$, $\bar y_2$,$\bar x_1$, $\bar x_2$ are ...
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1answer
1k views

Find the posterior distribution of θ

I have this problem Given the prior distribution is \begin{align}Pr(\theta=i)=\pi_i=\begin{cases} 0.5, & \text{for i=4}.\\ 0.3, & \text{for i=5}.\\ 0.2, & \text{for i=6}.\\ ...
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1answer
133 views

Finding Cumulative Distribution Function given two independent pdfs

The question is to find the Cumulative Distribution Function (cdf), of $W = X/Y$ given that X and Y are independent random variables and their pdfs are $f_x(x)=1,0\leq x\leq1$ and $f_y(y) = 1 , 0\leq ...
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1k views

Flip a fair coin repeatedly. What is the probability that the first sequence of heads is exactly two heads long?

Flip a fair coin repeatedly. What is the probability that the first sequence of heads is exactly two heads long? P(1st sequence = HH) = ? I am assuming that this is a conditional probability ...
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2answers
653 views

How many white balls are there in the box most probably?

There are n balls in a box. Some of them are white. A ball drawn from the box turns out to be white. How many white balls are there in the box most probably? Alright, well i know there are originally ...
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210 views

L2-Regularized\Penalized Logistic Regression

Suppose you have an $n$ dimensional data vector $x = (x_1, \ldots, x_n)$ and two classes $y = 0$ or $y = 1$. Assuming the dimensions of $x$ are conditionally independent given $y$, and that the ...
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1answer
127 views

Prove that either P(A∩B)=0 or else P(A'∩B')=0

Let E and F be independent with E = A∪B, and F = A∩B. Prove that either P(A∩B)=0 or else P(A'∩B')=0
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2answers
59 views

Correlation and independenc of two random variables

Let $X$ and $Y$ be random variable. Is it true that if $X$ and $Y$ are not correlation ( $ \mbox{corr}(X,Y) = 0$) then $X$ and $Y$ are independent?
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376 views

Who has a higher chance of drawing an easy question?

From a pool of N questions n are easy. Two people draw a question without returning it. Who has a higher chance of drawing an easy question?
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751 views

What is the probability C was the one with the wrong answer? [closed]

Students A, B and C each independently answer a question on a test. The probability of getting the correct answer is 0.9 for A, 0.7 for B and 0.4 for C. If two of them get the correct answer, what is ...
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4k views

Improbable vs Impossible?

I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossible ...
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25 views

distribution of spaced normal variables

If we have samples $x_1$, $x_2$, ...,$x_M$ in $R^N$ obtained as follows: 1) $x_1$ is drawn from a normal distribution $N(0,\Sigma)$ 2) $x_k$ ($k>1$) are also drawn from $N(0,\Sigma)$, but only in ...
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79 views

A limit technique in shifting the mean of normal distribution

This question arises from showing the existence of some strange admissible estimators in the theory of point estimate in statistics. Let $\varphi(x)=\exp(-{1\over 2}x^2)$, then show for any ...
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2answers
709 views

Why does the central limit theorem imply that the standard deviation approaches $\frac{\sigma}{\sqrt{n}}$?

According to the central limit theorem, if one takes random samples of size $n$ from a population of mean $\mu$ and standard deviation $\sigma$, then as $X$ gets large, $X$ approaches the normal ...
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1answer
703 views

Probability of secretary making 4 or more errors on a page [closed]

I have this problem, and I want to figure out how to do it, or at least figure out the subject that it deals with. A secretary who only does word processing makes $2$ errors per page when typing. ...
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1answer
853 views

normal approximation to binomial and poisson

My lecturer said that the following approximations are best when normal approximation to the binomial is best when np$\ge$5 and n(1-p)$\ge$5 normal approximation to the poisson when $\lambda$>20 ...
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1answer
262 views

Minimally Sufficient Statistic for Bivariate Distribution

Say you have: $f(x_i,y_i)=\lambda \mu \exp\{-\lambda x_i-\mu y_i\}$ However you only observe: $Z_i=\min (X_i,Y_i)~$ & $~U_i=\begin{cases}1~\text{if}~Z_i=X_i \\ 0~\text{if}~Z_i=Y_i\end{cases} $ ...
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2answers
375 views

Markov chains for beginners, how to think about them?

So this is what my book states: Random variables $X,Y, and Z$ are said to form a Markov chain in that order denoted $X\rightarrow Y \rightarrow Z$ if and only if: $p(x,y,z)=p(x)p(y|x)p(z|y) $ ...
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447 views

Calculate percentage with Days

I'm programming a projectscheduler and want to calculate the percentage of finishing by looping through all the subprojects and calculate their weight and how many percent is done by now and then ...
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655 views

Random process, stochastic process explained intuitively?

So I've read the definitions online and this is what I understood. $X(t)$ is a random process for $t>0$ and we can think of it as being a random variable at any given time $t=t_0$. For example, ...
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220 views

How to write conditional probability in terms of density function

I'm reading about statistical decision theory and started to wonder how would you write conditional probability in terms of density function? For example, if we have random variables $X$ and $Y$ then ...
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1answer
436 views

Probability of a normal distribution; more than, less than confusion.

You are interested in finding how many hours a person is willing to wait for a plane. It is found that the time people are willing to wait has a $μ = 5.2$ and a $σ = 1.1$. What is the probability a ...
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1answer
3k views

Probability of at least one event occuring through n trials.

How do I calculate the probability of an event happening at least once given a specific amount of trials? For example: There are 5 large black dogs, 2 large white dogs, 3 small black dogs, and 4 ...
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1answer
56 views

Mysterious failure to generate independent set of random variables

After getting my answer in this: Making a well conditioned orthonormal basis I am running into a problem which I do not understand. I have n dependent gaussian random variables that are related by a ...
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1answer
74 views

$P(|X - x| \leq |X - y|)$ with randvar $X$ on uniform distribution $[0,1]$ in terms of integers $x$ and $y$.

I'm sorry for this basic question, but my brain doesn't want to be functional at the moment, thanks for the help. What is the $P(|X - x| \leq |X - y|)$ with the random variable $X$ on uniform ...
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72 views

Probability of Runs of Heads of Length N [duplicate]

For example: $“THHTHTTHHHTHTHTTHHTHT”$ contains 1 run of heads of length 3, 2 runs of length 2, and 4 runs of length 1. Assuming $P(H) = p$ and $P(T) = (1-p)$, calculate (using properties such as ...
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107 views

Central moments of the wrapped normal distribution?

The wikipedia page defines the raw moments of the wrapped normal fairly succinctly: http://en.wikipedia.org/wiki/Wrapped_normal_distribution#Moments However, I'm struggling to find any literature on ...
3
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1answer
2k views

Baseball, batting average, and probability

A baseball player's batting average is equivalent to the probability he will get a hit for any given at-bat (at-bats don't include Errors, Walks, or HBP and a few other exceptions). So for a specific ...
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260 views

The third absolute central moment of the lognormal distribution?

Is there an analytic expression for $E|X-EX|^{3}$ of a lognormal distribution?
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73 views

$x_1, \cdots, x_n$ iid $N(0, \sigma^2)$ Find the value of c in order for it to have size $\alpha$

$x_1, \cdots, x_n$ iid $N(0, \sigma^2)$ and we're testing the hypothesis $H_0: \sigma^2= \sigma_{o}^{2}$ $H_1: \sigma^2= \sigma_{1}^{2}$ $\sigma_{0}^{2} > \sigma_{1}^{2}$ Im given the ...
3
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1answer
2k views

Expected squared prediction error

I'm reading about statistical decision theory and on one point in my book the author defines the expected squared prediction error by: $$EPE = E(Y-g(X))^2 = \int(y -g(x))^2Pr(dx, dy)$$ I like to ...
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69 views

$x_1, \cdots, x_n$ ~$N(\mu_{1}, \sigma^2)$ and $y_1, \cdots, y_n$ ~ $N(\mu_{2}, \sigma^{2})$ Find Restricted Likelihood Estimators

So the goal is to test the hypothesis $H_0: \mu_{1}=\mu_{2}$ $H_1: \mu_{1} \neq \mu_{2}$ I've already found the full likelihood function, and the unrestricted MLE's which is just the normal MLE's ...
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1answer
119 views

What does “$.05$ irrespective of the value of $\hat p$” mean?

I have a statistics question that says "What sample size would be required for the width of a $99$% confidence interval to be at most $.05$ irrespective of the value of $\hat p$?". But I'm not sure ...
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24 views

Establish that the power function of the test is equal to …

Establish that the power function of $\delta_{c}(x_1, \cdots, x_n)=I\{X_{(n)}\geq c\}$, that is $\pi_{c}(\theta)=P(\delta_{c}(x_1, \cdots, x_n)=1 | \theta)$ is equal to $\pi_{c}(\theta)= ...
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3answers
5k views

Guessing a number between 1 and 100 in 7 guesses or less.

This question was asked before but I have a lot of trouble understanding the answers given. Why is it that a number selected at random between 1 and 100 can be determined in 7 or less guesses by ...
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1answer
299 views

How to determine significance of a binomial test on a sample

I'm looking to determine the significance of the result of a binomial test of a sample of a population. Example: Given a group of 10,000 people, I ask 1,000 if they prefer iOS or android. 550 respond ...
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1answer
104 views

If $X_1, \dots , X_n \sim U[0,\theta]$ and iid. Verify that the distribution function of $X_n$

We know that $X_1, \dots, X_n$ are $ U[0,\theta ]$ and iid. I'm told to recall that the sufficient statistics and also maximum likelihood estimator of $\theta$ is $X_{(n)}=\max\{X_1, \dots, X_n\}$ ...
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1answer
229 views

Estimating sum of two gaussians using a single gaussian

The problem I have a sum of gaussians: $$ \Sigma(x) = p_1 e^{-\dfrac{1}{2}\left(\dfrac{x-u_1}{s_1}\right)^2} + p_2 e^{-\dfrac{1}{2}\left(\dfrac{x-u_2}{s_2}\right)^2} $$ which I'd like to ...
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81 views

Statistics - Covariance and variance question

Please fill in the intermediate steps $$\sum_{i=1}^nx_i(x_i-\bar x)=\sum_{i=1}^n(x_i-\bar x)^2$$ and $$\sum_{i=1}^nx_i(y_i-\bar y)=\sum_{i=1}^n(x_i-\bar x)(y_i-\bar y)$$