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.

learn more… | top users | synonyms

0
votes
1answer
413 views

Is the difference between the measured value and the theoretical value statistically significant?

I have a simulation in which I specify certain parameters, say $a$, $b$, and $c$. I have a model that I fit to the simulation results and find best fit parameters $a'$, $b'$, and $c'$. I do this many ...
0
votes
1answer
183 views

Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of ...
1
vote
1answer
37 views

Bias Method in monte carlo integration

This is from a proof in my monte carlo course. let $h$ be a smooth function, $T_n = h(\bar{X})$ $\mu = E(X)$ then by taylor expansion $E(T_n -\tau) = E[h(\bar{X} -h(\mu)] = E[\bar{X} - \mu]h'(\mu)+ ...
1
vote
0answers
29 views

What is the dot product of two randomly generated 0-mean unit-vectors?

Given pairs of random 0-mean unit vectors, what kind of distribution is generated by their dot products? Judging from a number of results I've generated myself, the distributions appear to be ...
0
votes
1answer
309 views

Calculating probability of winning Power Ball

Today I tried to calculate the probability of winning a jackpot in Power Ball California Lottery. There are 1-59 numbers from which you have to guess 5 out of 5 and 1-35 numbers from which you have to ...
0
votes
1answer
27 views

statistics - multivariate normal distn, variance and probability of event?

I have a multivariate Normal distribution defined by: μx = 360, μy = 280, μz = 180, σx = 40, σy = 34, σz = 48, and correlations of ρxy = −0.41, ρxz = −0.34, and ρyz = 0.47. I am required to find ...
2
votes
1answer
965 views

Derivative of double summation and dot notation?

I am trying to differentiate the following summation: $$ L(\mu, \tau_1, \ldots, \tau_i)= \sum_{i=1}^v \sum_{t=1}^{r_i} (y_{it}-\mu - \tau_i)^2 $$ $$\frac{dL}{d\mu} = y_{\cdot\cdot}-n\mu - ...
0
votes
1answer
31 views

Calculating Partitioned Matrices from subs

Say you have a matrix $A$ which is of size $P\times P$ and a number $Q < P$ can be used to take a partition of said matrix, where: $A_1$ is the upper-left sub matrix, with dimension $Q\times Q$, ...
0
votes
1answer
191 views

Pearson's chi square test mathematical notation

Is there any way to translate the statement below in full mathematical / symbolic notation without using any words? For example to rewrite it as a function f(A,E,x). If possible can someone show me ...
4
votes
2answers
345 views

Probability distribution of dot product?

Sorry if this is a basic question. I don't know much about statistics and the closest thing I found involved unit vectors, a case I don't think is easily generalizable to this problem. I have a ...
1
vote
1answer
103 views

How does SVD work?

Trying to find information, and, no-one seems to know the answers. I have a time-series, represented by $T = [0, 1, 1, 0, \ldots, n]$ the time series is then transformed into the Spectral results: ...
1
vote
0answers
41 views

Does this Gamma posterior make sense?

quick question about the form of a posterior distribution. Suppose that $\theta \sim Gamma(a, b)$ and that, given $\theta$, $Y$ has CDF $$F(Y\mid\theta) = 1 - e^{-\theta(e^y - 1)},\quad ...
1
vote
0answers
33 views

Which factors determine whether a set of variables are suitable for factor analysis?

Which factors determine whether a set of variables are suitable for factor analysis? I am looking as much for an explanation of the question as a tentative answer to it. So grateful for any help on ...
0
votes
1answer
60 views

What is the relationship between Multicollinearity and Standard error?

Can somebody give a clear explanation for the relationship between Multicollinearity and Standard error? I heard if multicollinearity exists, then the standard error goes up, but I don't understand ...
0
votes
1answer
48 views

Multicollinearity Check in statistics

What does that mean by "a linear dependence exists among three or even more variables" in multicollinearity? Can somebody explain it in plain English?
1
vote
0answers
85 views

What is the less confusing way to explain confidence intervals to a beginner

Let us say that you are back in high school and you have a friend who has missed class for a week. He needs information to be spoonfed to him, because its not his style to overthink. If you push for ...
0
votes
1answer
253 views

Why is $(x'x)^{-1}x' = x(x'x)^{-1}$

If $(AB)'=B'A'$ then $(x'x)^{-1}x'$ should be equal to $x((x'x)^{-1})'$ . However most econometrics textbooks say that this is equal to $x(x'x)^{-1}$ . What happened to the transpose of $(x'x)^{-1}$? ...
1
vote
1answer
72 views

How do you express the maximization step of EM algorithm in matrix formulation?

Specifically, I am interested in how the covariance matrix is calculated. In terms of dimensions of factors involved, let's say I am given some data set X of dimension m x d, covariance matrix S of ...
1
vote
1answer
136 views

Monte carlo estimation of maximum likelihood estimators

I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, ...
2
votes
1answer
66 views

Chebychev to get at least 90 %

Im am given a list of integers and asked to give the interval that contains at least 90 % of my values. Values : $62,56,72,83,66,77,62,71,50,58, 74,81,76,67,70,70,69,67,80,81, ...
0
votes
2answers
97 views

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$.

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$. I fell into this hole where I keep finding that $\sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i ...
0
votes
1answer
35 views

Degrees of Freedom of Counting Experiment

Lets say every hour I count the number of birds outside my window. I do this for 20 hours a day and record my data so each day is a row, and the first column is the number of intervals I counted zero ...
0
votes
1answer
86 views

Finding estimates of a Linear Regression Equation - R

I'm new to Statistics and R. I'm currently looking through a book called "Discovering Statistics using R". Although the book implies you don't need any statistical background, some of the content ...
0
votes
1answer
98 views

Delta Method and Sample Mean

Suppose I have the the sample mean from an iid sample $\bar{X}_n = \sum {X_i \over n}$ and define $\bar{E_n} = {2 \over n} \sum_i X_{2i}$. Now I would like to show that given convergence of ...
1
vote
1answer
75 views

Probability of Pressing Spacebar

What is the probability that the next key a user hits on a $51$-key keyboard (the keys being $26$ letters, one shift, one spacebar, one return, one backspace, $8$ regular punctuation keys, $13$ ...
0
votes
1answer
305 views

VC dimension for Rotatable Rectangles

It can be shown that VC dimension of rotatable rectangles is 7. The problem is I cannot understand how to approach the solution. So far I used bruteforce to solve this kind of problem, I was drawing ...
0
votes
1answer
161 views

Expected Value of a Lottery

You are deciding whether or not to enter a lottery. An entry ticket costs 50 cents. If you play, you either win exactly one of the prizes or nothing. The prize amounts and the chances of winning them ...
0
votes
2answers
510 views

In a distribution that is skewed by a few extreme outliers, what would be the best choice for a measure of central tendency?

According to my Professor, the answer to this question is the MEDIAN. However, wouldn't MODE be a better measure of central tendency? The mode is almost never affected by extreme outliers..please help ...
1
vote
1answer
55 views

Normal and standard distribution

There is some details i don't understand in my book, here goes; Let $X \sim N(\mu,\sigma^2)$ and $Z\sim N(0,1)$ we know that: $$F_X(x) = \int\limits_{-\infty}^{x} \frac{1}{\sigma ...
0
votes
1answer
99 views

Infinite Probability Space of Real Numbers

I have a question on my homework that I'm very befuddled about. Find a value of a constant $c$ such that the formula $p(n)=\frac{c}{3^n}$ defines a probability distribution on the set ...
1
vote
1answer
241 views

Intuition for Variance

Say $X$ is a random variable with a binomial distribution where $n = 3$ and $p = 0.5$, eg. the number of heads we get when we flip a coin three times. So this distribution has the values - $0$ with ...
2
votes
1answer
40 views

statistics and probability. Homework

I've been given the following problem: if I have 8 red and 10 blue marbles what is the probability that from 4 marbles that I took fom tha bag 2 are red? I missed some lessons. Any help would be ...
0
votes
2answers
153 views

What is the probability that a Chi-square distribution lies within 2 standard deviation of its mean?

Here I have an 8 degrees of freedom Chi-square distribution function $f(x)$ So by definition, $E(X)=8, Var(X)=2*8=16$. (Please guide me if this is wrong. We just started this chapter and there's ...
0
votes
1answer
65 views

Covariance of two random variables in a bivariate normal distribution

http://www.econstreams.com/bivariateproof.jpg Image uploaded to the link above. I'm just not seeing the connection between the 2nd equation on the left handside and the equation on the right. ...
1
vote
2answers
3k views

Average and variance of flipping a coin

A coin is flipped repeatedly with probability $p$ of landing on heads each flip. Calculate the average $\langle n\rangle$ and the variance $\sigma^2 = \langle n^2\rangle - \langle n\rangle^2$ of the ...
0
votes
1answer
78 views

Modeling viral growth

I'm trying to forecast the viral growth for a website I'm developing. I've arrived at some monthly growth rates I can expect per month, the difficulty is I'd like to model my growth per week instead. ...
1
vote
1answer
24 views

Using the negative binomial distr.

Tom "makes"(scores) a basketball shot with the probability $0.8$. He stops when he has made 10 scores. (A) what is the probability that he makes it in $13$ attempts so here i used the negative ...
1
vote
1answer
35 views

Correct terminology for confidence intervals

In Portuguese people speak of "intervalo unilateral à direita/bilateral" and Italians speak of "intervallo unilaterale destro/bilaterale" when referring to $P[S_1 ≤ \theta ]= \gamma$ and $P[S_1 ≤ ...
2
votes
2answers
31 views

How would I calculate the better performance based on percentages?

I've been looking at how the characteristics of web content relate to the number of views it receives. I looked at the percentage of content with that characteristic, and then the percentage of views ...
2
votes
1answer
181 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
0
votes
1answer
272 views

Covariance Matrix of zero mean complex vector

$$\textbf{f}=[f_1, f_2, f_3];\quad \textbf{g}=[g_1, g_2, g_3] $$ $f_1,f_2,f_3,g_1,g_2,g_3$ are all independent identically distributed zero mean complex random variable. h = elementwise wise ...
3
votes
2answers
787 views

What does $E[XY]$ mean?

Let's say I have two random variables, $X$ and $Y$. $X$ is the value of a fair die, $Y$ is the result of a coin flip, with heads being 1 and tails being 0. $E[X] = \sum_{k=1}^{6}{\frac{k}{6}} = ...
2
votes
1answer
42 views

Algebraically expanding this function

I have an algebra question. I am currently working on this function: $$\left(\frac{{e^{t\sqrt{3/n}}} -{e^{-t\sqrt{3/n}}}}{2t\sqrt{3/n}}\right)^n$$ Now this is the MGF of a function called $Z$ and I ...
3
votes
1answer
76 views

What am I doing wrong in calculating Fisher Information of Triangular Distribution?

I am trying to find Jeffrey's prior for the Triangular distribution which has the following probability density function: $$f(x\mid \theta) = \begin{cases} \dfrac{2x}{\theta} & : x ...
2
votes
2answers
63 views

Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of ...
2
votes
1answer
853 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
6
votes
2answers
649 views

How to find a confidence interval for a Maximum Likelihood Estimate

My cousin is at elementary school and every week is given a book by his teacher. He then reads it and returns it in time to get another one the next week. After a while we started noticing that he was ...
1
vote
1answer
4k views

Method of Moments Estimation over Uniform Distribution

this is my first time using this site so apologies if the formatting is unclear! Let ${X_1,\ldots, X_n}$ be a random sample from $\mathrm{Uniform}[\theta_1, \theta_2]$, i.e. the (continuous) uniform ...
2
votes
4answers
78 views

Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. ...
1
vote
1answer
28 views

Average deviation of uniform law

I compute the average deviation over a datasets of $n$ data in $[0,1]$ as $d=\sum_{i=1}^{n}\left|x_{i}-\mu\right|$ where $\mu$ is the usual average defined as $\mu=\sum_{i=1}^{n}x_{i}$. I know that ...