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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Finding the solution for a (differential) equation involving the pdf and cdf of a normal distribution

I have this (differential) equation: f(x)*x + F(x) = 1 where f(x) is a pdf and F(x) its corresponding cdf. Or similarly, F'(x)*x + F(x) = 1 I can calculate x numerically, but I want an analytical ...
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134 views

Quadratic Form MGF

I am not sure how to proceed with the question below. I know the theorem that states what the mgf is... I'm just not sure how to properly apply it. Essentially, from the question below, I know that ...
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1answer
263 views

Finding mean values, given the covariance matrix and some means

Suppose we have 4 normally distributed variables X1, X2, X3 and X4. The covariance matrix and mean values of X1 and X2 are given. Is there a way to determine the mean values of variables X3 and X4?
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32 views

What are the alternatives for this hypothesis test?

I know Type I errors are rejecting the null hypothesis when it is true. So I am 95% sure of the answer to this, just want to get some confirmation.
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64 views

Treat as one or two parameters?

Suppose you want to estimate the parameters $\alpha$ and $\beta$ of a $\text{gamma}(\alpha, \beta)$ distribution where we know that $\alpha = \beta$. Would you treat this as a distribution with one ...
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1k views

Mean and Variance of Methods of Moment Estimate and Maximum Likelihood Estimate of Normal Distribution.

I just wanna verify if I answered this question correctly. The following numbers are taken from a population having normal distribution with mean and variance : 5.3299 4.2537 3.1502 3.7032 ...
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1answer
135 views

Relation between repeat number in coin toss

I am trying to establish correlation between tossing of coins and occurring of repeats. Coin is flipped 10 time as follows: $${\rm H.T.H.H.H.T.H.T.T.T. }$$ After each repeat occurring ...
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4k views

Mean and Variance of Methods of Moment Estimate and Maximum Likelihood Estimate of Uniform Distribution.

Let $X_1, X_2,\ldots, X_n$ be i.i.d. uniform on $[0, \theta ]$. a. Find the method of moments estimate of $\theta$ and its mean and variance b. Find the MLE of $\theta$ and its mean and variance. ...
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80 views

Suppose that $e(t)$ is a white noise process, and consider the process $Y(t) = μ + e(t) - e(t-1)$

Suppose that $e(t)$ is a white noise process, and consider the process $Y(t) = μ + e(t) - e(t-1).$ Show that the process is stationary and compute its autocovariance function and ACF? Please ...
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284 views

t test question

Given a sample of 6 workers, three were randomly selected for working environment A and three were selected for working environment B. The observed response (the number of widgets produced) was for ...
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179 views

Standard normal distribution probabilities

Ok so I am having difficulty understand the concept behind standard normal distribution probabilities, in the questions I am getting a graph and a table FILLED with numbers, top header column has ...
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449 views

Find the Bayes estimate of $θ$

Can someone help me solve this out, please? Thanks a lot. Find the Bayes estimate of $\theta$ based on a single observation of $5$ from a distribution that is uniform on the interval 0 to $\theta$. ...
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62 views

About probability

Is a probability function in $\Omega{(a1, a2, a3)}$, find $P(a1)$ whether $P({a2, a3}) = 2P(a1)$. I know that $1 = a1 + a2 + a3$. from where I have to start ?
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104 views

How to convert the constants in a regression equation to constants in a linear equation

Hello dear mathematicians, I'm not entirely sure what to tag this question with since I'm new here but I hope some more experienced user can guide me. Here is my problem: I'm using an internal ...
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3answers
6k views

Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator. I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get ...
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1answer
162 views

Relation between power laws and stable distributions.

Who can explain me the relation between a power law and the stable distributions? Namely How i can represent a power law in terms of an alpha-stable distribution?
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2answers
107 views

Independence of Random Variables (kernel ICA)

In the paper Bach, F. R., & Jordan, M. I. (2002). Kernel Independent Component Analysis. Journal of Machine Learning Research, 3(1), 1-48. doi:10.1162/153244303768966085 I stumpled upon ...
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1answer
198 views

Kolmogorov-Smirnov test

Five observations from an underlying loss distribution are: $0.1,\ 0.2,\ 0.5,\ 0.7,\ 1.3$. Find the value of the Kolmogorov-Smirnov test statistic for test that the underlying distribution has p.d.f. ...
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1answer
76 views

Detecting lines and circles in x/y datasets

I'm trying to do some simple gesture recognition. Circles, I'm assuming, will simply be a set of points which are all within some specified distance range from center. Lines on the other hand, I'm not ...
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2answers
448 views

Multivariate Moment Generating Function

Let $X$ and $Y$ be two independent random variables both have Laplace distribution. What is the moment generating function of $U=X+Y$ and $V=X-Y$? Initially, I want to work out the $f_{U,V}(u,v)$, ...
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0answers
288 views

Bivariate normal distribution testing hypothesis problem

Let $(X_1,Y_1),\ldots,(X_n,Y_n)$, be a sample from a bi-variate normal distribution with zero means,variances $q_1^2$,$q_2^2$, and correlation $p$. How to Show that $$ r=\frac{\sum_i(X_i \cdot ...
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1answer
80 views

Which distribution can be abbreviated as “LD”?

Which distribution can be abbreviated as LD and which PDF is expressed as a formula with sum of erfc() functions? $$p(o)=\frac{1}{4\ell} ...
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151 views

Finding the covariance

Can you help me find the covariance of $\mathrm{Cov}(5X+3Y, 7X-Y)$ I've been looking up formulas all day and I can not find on that adds both x and y just constants. Thank you! $Mx=2, My=7, ...
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2answers
595 views

Why does Bayes Theorem work?

I have always wondered about the math behind Bayes Theorem because it looks really simple and seems like there's probably a simple explanation behind it. I don't understand the relationship between ...
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558 views

Bayes point estimate using mode of a Gamma posterior distribution

Let's say the posterior distribution of $\theta$ is Gamma with $$\alpha = 40, \qquad \beta = \frac{1}{0.5 + \sum_{i = 1}^{10} X_i}$$ What is the Bayes point estimate using the mode of the ...
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2answers
405 views

Show that $P$ is Countably additive problem.

Let $F$ be the field consisting of the finite and the cofinite ($A$ is cofinite if $A^c$ is finite) sets in an infinite and countable $\Omega$, and define $P$ on $F$ by taking $P(A)$ to be $0$ if $A$ ...
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2k views

How to apply Central Limit Theorem to Uniform Distribution to generate Normal Distrubution?

Suppose I have a simple uniform continuous "unit" distribution X: $$\begin{align*} \forall y \in \mathbb{R} \implies \\ y < 0 : & P(X < y) = 0 \\ y \in [0,1] : & P(X < y) = y \\ ...
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3answers
4k views

Correlation between two linear sums of random variables

I understand how to create random variables with a prespecified correlational structure using a Cholsesky decomposition. But I would like to be able to solve the inverse problem: Given random ...
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396 views

M.SE reputation distribution

What distribution does the reputation points per user follow on math.SE (or on entire stackexchange)? Is there a mathematical explanation/model of it?
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2answers
233 views

Independence and Events.

Let A, B and C are independent events. How am I supposed to prove that: A′, B′ and C′ are independent. A, B′ and C′ , are independent. A, B and C' are independend. This is my approach: for Nr 3. ...
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1answer
101 views

Three Independent Random Variables

$X_1,X_2,X_3$ be independent variables, and have marginal pdfs $f_{x_i}(x_i)=c_ix_i^ie^{-x_i}$, $x_i>0$. Let $Y_1=X_1/(X_1+X_2+X_3),Y_2=X_2/(X_1+X_2+X_3),Y_3=X_1+X_2+X_3$. How can I know if ...
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99 views

Independent Events [duplicate]

Possible Duplicate: Independence and Events. Let $A$, $B$ and $C$ are independent events. How am I supposed to prove that: $A'$ , $B'$ and $C'$ are independent and $A, ...
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2answers
1k views

What is the distribution of the euclidean distance between two normally distributed random variables with non-zero means?

Assuming two uncorrelated random variable (RVs) with Gaussian distributions $x\sim N(m_1,s)$ and $y\sim N(m_2,s)$, so with non-zero mean and same variance, what is the distribution of $z=\sqrt{(x^2 + ...
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101 views

standard deviation computation

If I currently have a sum of N elements (I don't have the elements them self), their mean and the corresponding standard deviation. Later on I receive M (known) other elements. I want to update the ...
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350 views

Need help finding value (or contingency table) for Chi-squared critical value.

I need help finding value (or contingency table) for Chi-squared critical value at 95% significance level when degrees of freedom is 58. I have calculated the chi-square calculated value, and I need ...
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1answer
87 views

Homework: calculations for $I(\sigma^2)$

This is a solution to 1 of my tutorial questions, anybody got any clue as to the step: $\frac{1}{nI(\sigma^2)} = \frac{2\sigma^2}{n}$ ? Thanks.
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1answer
78 views

Finding distribution $W$

Suppose $X_1,\ldots,X_m\sim E(\mu,\sigma_1)$, $Y_1,\ldots,X_n\sim E(\mu,\sigma_2)$ are two independent random sample. If $W=\min(X_{(1)},Y_{(1)})$, how can find distribution $W$? Note: $X_{(1)}$ is ...
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240 views

Expectation Maximization algorithm

I am having problems finding a well thought out complete explanation of expectation maximization. Does anyone have a best source for someone completely new to this stuff?
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2answers
183 views

Working out the “best” score based on quantity and ratio

I'm not an expert with mathematics so I hope I'm posting in the right place! I've got a lot of products which can be rated good, bad and OK by a user. What I'm having trouble with is finding which is ...
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3answers
231 views

Why doesn't integrating infinitesimally small likelihoods work in this sense?

We know that something is going to happen after $x$ amount of time, but the exact time at which the event occurs is random within $x$ time. (Like, say we did it a bunch of times where it happened in ...
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4answers
6k views

What is the purpose of the standard deviation?

I don't have any knowledge of statistics beyond high school common sense. Why is the standard deviation usually seen in combinatorics textbooks, and why is the standard deviation defined ...
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177 views

Panjer Recursion and Expectation

Given a Panjer Recursion set up, with the usual properties, and supposing now that $N$ has a Poisson distribution with mean $\lambda$. How can we derive a recursion for $E(S^k)$ where $S$ be the ...
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1answer
2k views

Likelihood Ratio Test for Linear Regression

I apologize for the image I am posting below. I am new to StackExchange and I am not yet familiar with the MathJaX equations, so I took a screenshot. Here is my question: Let the independent random ...
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1answer
71 views

Normal Distribution Properties

Hi all i have this question in which i don't really understand the line of reasoning. Given rvs X ∼ N(0, 1), Y ∼ N(0, 4), is P(X > 3) < P(Y < –6)? The reasoning is : P(Y < –6) = P(Y > 6) ...
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155 views

The Central Limit Theorem

Suppose $X_1, \dots, X_{20}$ are i.i.d random variables with pdf $f(x) = 2x, 0 < x < 1$. Find $P(S < 10)$ where $S = X_1+ \cdots + X_{20}$. So find $E(X)$ and $\text{Var}(X)$. Then $S$ has ...
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135 views

'Odd' Odd ratio question

Assuming Intervention group to be mothers having c-section and control group to be mothers having natural birth. Also, the numerical data is as follows: ...
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45 views

Adequate Trial Size Calculation.

Firstly, thanks for answering my last question. I appreciate the help and effort put in. I have another question to put forth. Again, it is related to Medical Statistics. There are one third of ...
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1answer
847 views

UMVUE of parameter $\frac{1}{1+\lambda}$

suppose $X_1,X_2,X_3$ are a random sample of exponential distribution with parameter $\lambda$. how can find UMVUE parameter $\frac{1}{1+\lambda}$. note: $(T=\displaystyle\sum_{i=1}^{3}X_i,\ ...
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2answers
6k views

Error propagation on weighted mean

I understand that, if errors are random and independent, the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the ...
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320 views

Expected value for the number of goals in a game

I'm trying to use odds data from bookmakers to estimate the expected number of goals in a game. We have these known facts: P(o4.5) = 0.573 P(o5.5) = 0.458 P(o6.5) = 0.279 P(o4.5) is the ...