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

5
votes
3answers
149 views

How to Make a PDF 'Look' Uniform?

Let $X$ be a normally-distributed random variable with mean zero and variance $\sigma^2$: $X \sim N(0,\sigma^2)$. Let $Y$ be a mapping from $X$ onto the interval $(0,1)$ using the sigmoid function: ...
5
votes
2answers
116 views

Convergence in probability

If $X_1, X_2, \ldots$ converge in probability to a constant $c$, then does $1-X_1, 1-X_2, \ldots$ converge in probability to $1-c$? Is there a way to show this is true / is there an already existent ...
5
votes
4answers
436 views

How to find unique multisets of n naturals of a given domain and their numbers?

Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set ...
5
votes
2answers
115 views

Functions whose derivatives can be written as a function of themself?

What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$? If $f$ is given, can $g$ be solved in terms of the ...
5
votes
3answers
80 views

Bernoulli Random Variables and Variance

The question is: Suppose $Z_1, Z_2, \ldots $ are iid $\operatorname{Bernoulli}\left(\frac{1}{2}\right)$ and let $S_n = Z_1 + \ldots +Z_n$. Let $T$ denote the smallest $n$ such that $S_n = 3$. ...
5
votes
2answers
228 views

Statistics and confidence - intervals

An account on server A is more expensive than an account on server B. However, server A is faster. To see whether it's optimal to go with the faster but more expensive server, a manager needs to ...
5
votes
1answer
541 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 ...
5
votes
1answer
2k 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 $\rho \in [-\frac 12,1]$, ...
5
votes
2answers
3k views

Defective items probability question.

Hi I'm working with probability as part of an engineering course, and I'm struggling with the following tutorial question: Components of a certain type are shipped to a supplier in batches of ten. ...
5
votes
2answers
462 views

What is the relationship between the Poisson Distribution and the Monte Carlo Fallacy?

Gravity's Rainbow has this long passage about the Poisson distribution. Since Pynchon's education included a serious dose of mathematics, and his novels include many references to mathematics, I ...
5
votes
1answer
3k views

X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$

X, Y are independent standard normal random variables, what is the distribution of $$ \frac{X}{X+Y} $$ Could anyone help me with this? Thanks. I have worked the problem by multivariable ...
5
votes
1answer
522 views

Probability of Monkey typing keyboard

A monkey types at a 26-letter keyboard with one key corresponding to each of the lower-case English letters. Each keystroke is chosen independently and uniformly at random from the 26 possibilities. ...
5
votes
1answer
458 views

Expected Value of a Continuous Random Variable

I've been reviewing my probability and statistics book and just got up to continuous distributions. The book defines the expected value of a continuous random variable as: $E[H(X)] = ...
5
votes
2answers
290 views

If $X_1, …, X_n$ are Exp($\lambda$) random variables, what is the best unbiased estimator of $e^{-\lambda}$?

Let $X_1, ..., X_n$ be random variables with pdf $$\frac 1 \lambda e^{-x / \lambda} I(x > 0).$$ The goal is to find the best unbiased estimator of $h(\lambda) = e^{-\lambda}$ (incidentally, this ...
5
votes
4answers
2k views

A good book on Statistical Inference?

Anyone can suggest me one or more good books on Statistical Inference (estimators, UMVU estimators, hypotesis testing, UMP test, interval estimators, ANOVA one-way and two-way...) based on rigorous ...
5
votes
3answers
3k views

What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
5
votes
2answers
391 views

expectation of $ \left(\sum_{i=1}^n {x_i} \right)^2 $

If $x_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $x_i$'s are mutually independent, what is the expectation of $\left(\sum_{i=1}^n {x_i} \right)^2$ in terms of $n$ and ...
5
votes
2answers
4k views

Calculating a sample size based on a confidence level

It's been a while since my last statistics class... I have 404 files that went through some automated generation process. I would like to manually verify some of them to make sure that their data is ...
5
votes
2answers
63 views

$X-Y$ equivalent in distribution to $0$?

If $X$ is equal to $Y$ in distribution, is it equivalent to $X-Y$ which is equivalent in distribution to $0$?
5
votes
2answers
293 views

The Birthday Problem

I've been reading about the birthday problem which, as I'm sure many of you will know, is a statistical problem which aims at finding out the how many people you would need in a random group to be ...
5
votes
2answers
119 views

Highest points in heart cycle graph

I'm making an application that reads the heart cycle from a device, and I've aimed to get this image: Now, I need to get the highest points that appear in every cycle in order to calculate the ...
5
votes
3answers
229 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
5
votes
1answer
162 views

Conditioning on an event with probability close to one

Let $(\Omega,\mathcal{F},P)$ be a probability space. If $A\in\cal F$ is an event with $P(A)=1$, then $$ P_{\mid A}(B)=P(B\mid A)=\frac{P(B\cap A)}{P(A)}=P(B),\quad B\in\cal F. $$ I wonder if something ...
5
votes
2answers
1k views

Probability/Combinatorics Problem. A closet containing n pairs of shoes.

A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that the chosen shoes will contain no matching pair? I have tried thinking about this ...
5
votes
2answers
89 views

Greatest common denominator of measurements

In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ ...
5
votes
1answer
144 views

Proof of $ \text{Var}\,\left(\sum_{i=1}^{n}g(X_i)\right)=n\left(\text{Var}\,g(X_1)\right).$

I have a question about part of a proof of a Lemma in a book (Casella's Statistical Inference) I'm reading. This it how it goes. Let $X_1, \cdots ,X_n$ are a random sample from a population and ...
5
votes
1answer
112 views

Expected overlap

Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large ...
5
votes
1answer
170 views

Statistics formula for wifi positioning.

Assuming I have $3$ access point namely: $AC_1$, $AC_2$ and $AC_3$ and I want to know my location using this access point and a mobile device that will get signal from the access points. First thing ...
5
votes
1answer
103 views

Calculating the average of a possibly infinite “compound” length

Sorry for the ambiguous title, I couldn't find a good word to describe my problem. So here is my problem: You are a player, and you have a dice. You have N number of throws available then you can't ...
5
votes
1answer
186 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
5
votes
2answers
752 views

The probability of a drunk person/random walk

A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$. a) After 10 steps, what is the probability that he has moved 2 steps ...
5
votes
2answers
80 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 ...
5
votes
2answers
148 views

Why $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

I'm reviewing probability and statistics.The textbooks said that if the sampled population is infinite, then $$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$$ I'm curious about how does this ...
5
votes
2answers
487 views

Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)

Edited: My question is related to a tutorial I was reading. The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
5
votes
2answers
395 views

Expectation and Variance of Ratio Estimator

Let $X$ and $Y$ be positive random variables such that $$E(Y\mid X)= aX $$ $$\operatorname{Var}(Y\mid X) = b^2X^2 $$ $$a,b > 0 \text{ are constants}.$$ Let $R = ...
5
votes
2answers
368 views

Help with question on joint Gaussian distribution

Does anyone know how to start this question? Let random vectors $x,u,v$ have joint Gaussian distribution, and $u,v$ be independent. Show that $E(x|u,v)=E(x|u)+E(x|v)-E(x)$.
5
votes
1answer
182 views

Finding the joint distribution of $X_{1:n}$ and $\overline{X}$

I need to show that, given a random sample of independent variables $X_1, ... , X_n$, each following a distribution EXP($\theta$,$\eta$), that is, ...
5
votes
1answer
232 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
5
votes
1answer
95 views

Calculating success chance from algorithm

Not super sure this is the right *exchange for this question, but here we go. Let's say I'm writing a game, and in this game the player may attack another unit. The chance of hitting is an "opposed ...
5
votes
2answers
204 views

Working out minimum sample size

I have just started a course in statistics and have some general questions that have arisen trying to solve the following question: A survey organisation wants to take a simple random sample in order ...
5
votes
1answer
98 views

Godambe estimating equation (Proof)

Let $Y_1, \ldots ,Y_n$ be iid with density $f(y;\theta)$. We assume that $\dfrac\partial{\partial\theta}\log f(y ; \theta)$ and $\dfrac{\partial^2}{\partial\theta^2}\log f(y ; \theta)$ exist for all ...
5
votes
1answer
631 views

Finding the MLE for parameter $\theta$ from distribution of the form $e^{-|x-\theta|}$

this is my first post so I apologize if the formatting is a little rocky. I'm currently going through "Probability and Statistics" 4th ed by DeGroot/Schervish, and I was wondering if somebody could ...
5
votes
1answer
91 views

How to prove that second derivative of $\log\big(\int_{-\infty}^x e^{\frac{-t^2}2} dt\big)$ is $>-1$?

Let $\Phi(x)=\int_{-\infty}^x e^{\frac{-t^2}2} dt$. How can I prove that $$\left[\frac{e^{\frac{-x^2}2}}{\Phi(x)}\right]'>-1?$$ I could prove that its $lim$ at $-\infty$ is $-1$ and at $\infty$ it ...
5
votes
1answer
485 views

Calculation of the moments using Hypergeometric distribution

Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$. Define ...
5
votes
1answer
129 views

Poisson Process - Courts

IITK sports facility has $4$ tennis courts. Players arrive at the courts at a Poisson rate of one pair per $10$ min and use a court for an exponentially distributed time with mean $40$ min. Suppose ...
5
votes
1answer
110 views

Help with convergence in distribution

$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$ Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity? I let $Z = ...
5
votes
1answer
149 views

Show that the posterior density of ($\mu$, $\tau$) is equal to $f(\mu, \tau | x_1, …, x_n) = f(\mu| \tau, x_1,…,x_n)f(\tau|x_1,…x_n)$

Here is the full problem: Let $X_1,...,X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution. Let $\tau = \sigma^{-2}$, so we can write the distribution as $N(\mu,\tau^{-1})$. Suppose the ...
5
votes
2answers
672 views

What's the expected value of average absolute deviation from the mean of k randomly picked numbers?

Say we have to randomly pick k integral numbers out of n. The numbers are from the range < a; b >. What is the expected value of average absolute deviation from the mean for that random subset of ...
5
votes
1answer
69 views

Getting a single-value estimation of trust in a computed mean

Suppose I have a number N of independent ratings of a given item, where each rating is an integer between 1 and 7 (inclusive). For simplicity sake, let us assume the ratings are normally distributed, ...
5
votes
0answers
66 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...