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

2
votes
1answer
78 views

Regression with arbitrary norm

I have a function $f(X,Y,Z)$, which I know is polynomial of degree $3$. I have a set of samples $(X_i,Y_i,Z_i)$ and corresponding values of $f$. My task is to find (the best approximation of) the ...
1
vote
0answers
95 views

Compute significance of Kendall tau-b?

I have so-far tried all ways of computing kendall tau significance (where there are ties) described here. However, none of them works good, even for relatively large vectors. I think the problem is ...
2
votes
2answers
65 views

Creating indices

Is there a "proper" formula for creating indices? I need to compute series of numbers into a KPI that can be tracked over time. Example dataset is like this: ...
3
votes
2answers
476 views

The probability for guessing a dynamic value (one that changes for each wrong guess)

Please correct me if I am wrong... For a random number X between 1 and 100, the probability for guessing X correctly would be 1/100 For 2 random numbers X and Y between 1 and 100, the probability ...
1
vote
0answers
140 views

Measuring a mean variance for some number of objects observed per trial for multiple trials

I'm running a bunch of trials, $T$, and the outcome of each trial is some number of objects $k_i$ for $i = [1, T]$. I would like to say something about the average "spread" in terms of the number of ...
1
vote
1answer
396 views

The distribution when combining two samples together?

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
5
votes
2answers
96 views

Disprove independence of vector of Gaussians by independence of marginals

If we have three random variables $X,Y,Z$, then if $X$ and $Z$ are independent, and $Y$ and $Z$ are independent, it doesn't follow that $Z$ is independent of the vector $(X,Y)$. There is a simple ...
4
votes
1answer
195 views

Confidence interval for sum of random subsequence generated by coin tossing

This question is related to Sum of random subsequence generated by coin tossing. Here is the corresponding problem description as given by Memming: Let $(\pi_1, \pi_2, \cdots)$ be an infinite ...
1
vote
1answer
181 views

Numerology and the possible combinations

I was looking into numerology and it's pretty interesting to see how your name and birthdate are broken down into numbers and the categories that are used to assign numbers to them. My breakdown is ...
4
votes
1answer
94 views

Bayesian inference

I'm a bit confused with arranging the Bayes equation to update probability. Say, I have the following data: $P(\text{blue birds in the whole study area}) = 0.16$; $P(\text{all except blue colored ...
2
votes
1answer
151 views

Stein's Paradox and James Stein Estimator

I am trying to figure the intermediate steps in the proof of the Stein's paradox. How does it go from the left to the right? $$\frac{\partial}{\partial Y_i} \left\{\frac{Y_i }{\sum_j Y_j^2}\right\} = ...
2
votes
1answer
264 views

Optimization, Gradients, and Multivariate Data

I would like to learn gradient based optimization for multivariate data. For example, assume the data I have is $X = (x_0, ..., x_n)$ where $x_i$ are some random variables and $f$ a function ...
1
vote
1answer
58 views

What's the name of this phemonenon in random walks?

Given a random walker on the number line that starts at 0 that has a 50% chance of going 1 unit in either direction every step, the walker will tend to stay on one side of the line for a while before ...
1
vote
1answer
34 views

Determining rating scale based off the options

I'm trying to rate each value with a scale. How can I determine the rest of the ratings below? less weight is better rating scale from 0-10. 10 being the best 2.32 lbs - 10 2.47 lbs 2.55 lbs 2.87 ...
0
votes
0answers
316 views

What is an observable random variable?

On Wikipedia, a statistic is defined to be an observable random variable ( http://en.wikipedia.org/wiki/Statistic ). However, neither of the terms statistic and observable is formally defined in the ...
0
votes
0answers
28 views

Get a representative vector from a large set, and compare it with samples.

We're a small team of programmers and we're triying to solve a little problem, but we think we need some advices from professional mathematicians. We want to know if a picture of a card is an ...
0
votes
2answers
780 views

binomial distribution with two random variables

my teacher said that when you have two random variables X and Y, both are binomially distributed, then X-Y can never be binomail distribued? Why? I recall he mentioned something because it only takes ...
2
votes
0answers
93 views

Statistical Data Reconstruction

As I read an introductory statistics book, a question struck me which am curious to find a good answer for. Background The author teaches that statistical classification (understand this to mean ...
1
vote
0answers
138 views

How can I create a good subset of data proportional to the full data set

I am programming a neural network and I need to use some data to train it. I am following the steps of an experiment which tried to predict direction of stock price index movement using neural ...
0
votes
0answers
62 views

Define a problem using chernoff bounds

We are preparing this for an exam. Given the division of a plane into a number of regions of different sizes. We would like to find, or guess, which is the biggest region, by doing the following. We ...
1
vote
1answer
404 views

How do I find the frequency or percentage chance for two independent timed events to happen simultaneously?

If I have two events that occur on a specific interval (one every 8 seconds, the other every 200 milliseconds) but were not started synchronously, how can I calculate the frequency with which these ...
3
votes
1answer
62 views

Can a random function of a normal variable be independent of it

I have two normal variables $X,Y$. Is it possible to have $X = f(Y,Z)$ for some non-trivial function $f$, an independent normal variable $Z$ and that $X$ will be independent of $Y$? Thanks.
1
vote
1answer
107 views

How to model a system for tracking a person using kalman filter?

I need to model a system for human motion. The following link shows for to build a system for a plane. I am currently reading the documentation for a kalman filter library ...
2
votes
3answers
4k views

Inverse of random variable

Let's say we have a random variable $X$ of which the distribution is unknown. Now there are these general rules like $E[X + Y] = E[X] + E[Y]$ etc. But what if we would define $ \quad Y = \dfrac{1}{X} ...
2
votes
2answers
627 views

Is the maximum-likelihood estimator always biased?

To avoid answers too complicated for me to understand/to clear notation, I'm describing quickly the setting in which everything takes place: In our intro statistic lecture the following we said ...
2
votes
0answers
60 views

Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

In our intro statistic lecture the following we said that the following components made up an estimation problem an at most countable space $\mathcal{X}$ of all possible samples we can observe a ...
1
vote
0answers
106 views

identify if exponential distribution is a good model

I must identify if for a specific data set, the exponential distribution is a good model. Without using any plot. So what i was thinking is, to see if in the data set the rate of events is constant. ...
7
votes
2answers
185 views

What is the T-distribution, and what is it used for?

(I'll post my own answer to this, but don't hesitate to post your own!) Student's t-distribution, or T-distribution, was introduced in 1908 by William Sealey Gossett writing under the pseudonym ...
7
votes
1answer
1k views

Expected number of card draws to get all 4 suits

You have a standard 52 card deck, with 13 cards of each of the 4 suits (Hearts, Diamonds, Spades, Clubs). What is the expected number of cards you have to draw from the deck until you have all 4 suits ...
2
votes
2answers
285 views

Closed form for coefficients in Multiple Regression model

I want to find $\hat{\beta}$ in ordinary least squares s.t. $\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X_1 + \cdots + \hat{\beta}_n X_n $. I know the way to do this is through the normal equation using ...
2
votes
2answers
4k views

How can I tell if dice are biased or unbiased, given a number of trials?

If I'm given the outcome of a number of dice rolls (say, 5 twos, 8 threes, etc), is there a way to assign a probability that the dice are biased or unbiased? If so, how? Or alternatively, how can I ...
2
votes
1answer
4k views

Equations For Quadratic Regression

Does anyone know the specific equations for the three parameters in a least-squares quadratic regression? I'm looking for something like $\beta_1=,\beta_2=,\beta_3=$ for each of ...
1
vote
1answer
277 views

How to deal with non random data in statistical analysis?

I have a set of monthly water quality data, and I want to use them in a few statistical analysis (such as finding distribution or using in copula models) which require random variables as input. I ...
1
vote
0answers
38 views

How to find BUE of N? where $X_i$ iid $ P(X_i=j)=1/N$, for $j=1, \cdots, N$

Let $X_1, \cdots, X_n$ be iid with $P(X_i=j)=1/N$, for $j=1, \cdots, N$, where $N$ is an unknown positive integer. I would guess that the order statistics $X_{(n)}$ is the complete sufficient ...
0
votes
1answer
123 views

Find probability to reject $H_0$ when mean is given

The exercise is: Assume that the population mean is actually 110 grams and that the distribution is normal with standard deviation 4 grams. In a z test of $H_0$: u = 113 against $H_a$: u < 113 ...
-1
votes
3answers
473 views

Probability of rejecting faulty Bearings

A ball bearing diameter is $3.00 \pm 0.01$. The Mean and Standard Deviation are given (you can assume them to be any value). Using normal distribution, find the probability of a faulty bearing.
3
votes
2answers
809 views

Random Variable Probability Russian Roulette

I am very confused on this one. Any help in how to solve it and what probability rule to use would be appreciated. Consider the situation in which a person played Russian roulette (one bullet, 6 ...
0
votes
2answers
124 views

Mean of a possible MLE ,cant figure out the distribution of the sum of $x^{2}$

I would like to know: Is $\beta$ a MLE? If yes what is the mean of it: $E(\beta)=$? given: $x$ is a random variable $$f(x)=\sqrt{\frac{2}{\pi \theta^2}}\exp ...
4
votes
3answers
508 views

Does the Least Squares Regression Method work for any line type?

I recently learned how to apply the least squares method to do linear regression. I also understand that it can be used for quadratic regression, by minimizing the error for three variables, two ...
0
votes
1answer
126 views

Frequency table in A-Level Statistic book S1

I am confused with the answer given in the A-Level book S1. Please see the following question and please tell me why answer is 38 and not 24. Is it misprinting or am I missing out something? ...
-1
votes
1answer
190 views

Joint PDF of Solid Triangle,$ x+y\leq 1$ find $Px,y(x|y)$ for solid triangle?

$x+y\leq 1$ find $Px,y(x|y)$ for solid triangle?
2
votes
2answers
264 views

Algebraic manipulation of probability distributions

If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$, then $$ \frac{n - 1}{\sigma^2}S^2 \sim \chi^2_{n - 1} $$ where $S^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i^2- \bar{x})^2$, and there's a direct relationship ...
2
votes
0answers
88 views

Regressing $Y$ back on the residuals

Suppose I have the linear regression model $ \hat{y_i} = a + b x_i $ for $a,b$ obtained via OLS. How does one regress $y$ back on the residuals $\hat{e}_i = y_i - \hat{y}_i$? If we write $ ...
1
vote
0answers
48 views

Gaussian prior from feature to input space

if I have Gaussian prior ($\exp\left(\dfrac{-\sum_i w_i}{2\gamma^2}\right)$) on my weights in a linear classifier, how can I transform this so I can apply it for my kernel parameters $\alpha$? I have ...
2
votes
2answers
74 views

Determining If 2D Collection of Points Are Random Or Not

Please forgive the lack of depth in respect of my mathematical vocabulary. I have searched quite a bit in order to determine the answers for myself, however, if you don't know what ...
18
votes
3answers
5k views

Intuitive explanation of a definition of the Fisher information

I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this: $$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
1
vote
1answer
678 views

Dependent Bernoulli trials

The probability of a sequence of n independent Bernoulli trials can be easily expressed as $$p(x_1,...,x_n|p_1,...,p_n)=\prod_{i=1}^np_i^{x_i}(1-p_i)^{1-x_i}$$ but what if the trials are not ...
2
votes
0answers
218 views

What is the computational complexity of the EM algorithm?

In general, and more specifically for Bernoulli mixture model (aka Latent Class Analysis).
5
votes
1answer
3k views

pairwise correlation of three random variables

Assume three random variables have all equal pairwise correlation. What are the possible values of this correlation? Can all of these values be achieved? The solution says $\rho \in [-\frac 12,1]$, ...
4
votes
2answers
278 views

Restrictions on a bounded variable's mean and standard deviation

Given a random variable whose values are between $0$ and $1$, it is not difficult to see that the mean, or the expected value, is between $0$ and $1$, and the standard deviation is between $0$ and ...