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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I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll?
I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll?
I am not sure how to set this one up. This one is not homework, by the way, but a question I am making up ...
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1answer
787 views
What's the difference between Rao-Blackwell Theorem and Lehmann-Scheffé Theorem?
I know that the Rao-Blackwell theorem states that an unbiased estimator given a sufficient statistic will yield the best unbiased estimator. Is the only difference between Lehmann-Scheffé and ...
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4answers
385 views
What does it mean when a statistician says I’m 90% confident that the mean of the population is between 1 and 9?
Does that mean if I draw samples from the population that 90% of the time I'll get a number between 1 and 9?
Added: assume normal distribution for the population.
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499 views
What is the expected number of runs of same color in a standard deck of cards?
Standard deck has 52 cards, 26 Red and 26 Black. A run is a maximum contiguous block of cards, which has the same color.
Eg.
(R,B,R,B,...,R,B) has 52 runs.
(R,R,R,...,R,B,B,B,...,B) has 2 runs.
...
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123 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 ...
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263 views
What is degree of freedom in statistics?
In statistics, degree of freedom is widely used in regression analysis, ANOVA and so on. But, what is degree of freedom ?
Wikipedia said that
The number of degrees of freedom is the number ...
6
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1answer
936 views
easy to implement method to fit a power function (regression)
I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too.
I'm using in general financial data in my project, which ...
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483 views
What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?
If I have a fair die and throw it until I get a run of 1,2,3,4,5,6 in order, how many times on average must I throw the dice?
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4answers
98 views
What is a good measure of “controversy”, given a support score and opposition score?
Suppose I have a topic or discussion, and a number of "support" and "opposition" points on each side (You can also think of them as "upvotes" and "downvotes") and I want to calculate a score of how ...
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1answer
72 views
Monte-Carlo for the Wasserstein metric
Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let
$$
\Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
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2answers
180 views
Intuitive explanation of a definition of the Fisher information
I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this:
$$I(\theta)=E_\theta\left[-\frac{\partial^2 ...
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2answers
330 views
Bag with infinite number of colored balls
Consider a situation with a bag with infinity number of balls. Each ball is of some color. Number of colors is finite but it is not known. Balls are drawn from the bag one by one and checked for the ...
6
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1answer
148 views
Does “50/50 chance of.. . ” convey information?
I distinctly remember the professor in the undergrad introductory systems & control course saying that "when weather forecasters say there's a 50% chance of precipitation, they are conveying no ...
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1answer
2k views
Average percent increase not equal to total percent increase?
I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway.
What's an easy way to show that the average percentage increase of n ...
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1answer
295 views
License Plate Statistics
California issues license plates in numeric order (if we turn the letters into numbers). I have fun noticing the latest plate I have seen. I am interested in what you can derive from a series of ...
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1answer
451 views
Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process
I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
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2answers
184 views
Might such a sequence of mathematical expectations be able to predict uncertain events?
This question might sound a little bit mystical, but it seemed like an interesting idea, so I am posting it here. Despite the title, I know it probably does not work miracles, but here goes anyway.
I ...
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109 views
Estimate the size of a set from which a sample has been equiprobably drawn?
Here is the problem I'm trying to solve:
In order to send spam, a spammer generates fake nicknames, by picking random girl names (and appending a random number to it). I suppose it randomly and ...
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1answer
775 views
Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?
I have a covariance matrix:
$\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$
According to ...
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1answer
444 views
Symmetric matrix decomposition with orthonormal basis of non-eigenvectors
I like to understand the following transformation found in documentation for deriving Kalman filter.
Abstract Formulation: Given 2 symmetric matrices $A$ ,$B$ $\in$ $\mathbb R^{3,3}$ with $A \ne B$ ...
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150 views
Empirical distribution vs. the true one: How fast $KL( \hat{P}_n || Q)$ converges to $KL( P || Q)$?
Let $X_1,X_2,\dots$ be i.i.d. samples drawn from a discrete space $\mathcal{X}$ according to probability distribution $P$, and denote the resulting empirical distribution based on n samples by ...
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1answer
175 views
How to find a flat using game theory?
I had the idea that maybe probability/game theory knowledge helps finding a flat more systematically. I assume that I have some online offers with number parameters:
prize
size (square meters)
...
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1answer
166 views
If W is a random matrix with variance $\mathbb{E}[W W^{T}]$, what's $\mathbb{E}[W^{T} P W]$?
I know quite a few identities about quadratic forms of random vectors, but I'm having difficulty coaxing something out of this quadratic form of random matrices. Suppose I know $\mathbb{E}[W W^{T}]$ ...
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364 views
“How many public playgrounds exist in the United States?” How to answer using statistics and probability
I have a goal of estimating how many public playgrounds exist in the United States.
There are many methods of gathering real data about playgrounds, but, unfortunately, there is no single authority ...
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204 views
Does this calculation have a name, or a generic formulation?
Background
I would appreciate help in identifying / explaining this operation:
To calculate each of the $n$ values of $f(\Phi)$:
sample from the distribution of each of $i$ parameters, $\phi_i$
...
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2answers
981 views
Is it unheard of to say that you like math but hate proofs? [closed]
I have enjoyed math throughout my years of education (now a first year math student in a post-secondary institute) and have done well--relative to the amount of work I put in--and concepts learned ...
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5answers
467 views
Can an event be possible if its probability is zero?
Consider a computer program that generates any random number between 0 and 1(exclusive). There are infinitely many numbers between 0 and 1. So the probability that the random-number generate the same ...
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4answers
1k views
Unbiased Estimator for a Uniform Variable Support
Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $. Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be ...
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4answers
1k 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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2answers
496 views
Uniform distribution with probability density function. Find the value of $k$.
For a random sample $X_1,X_2,...X_n$ from a uniform $[0,\Theta]$ distribution, with probability density function
$$f(x;\Theta) = \left\{ \begin{array} \ \frac{1}{\Theta} & 0\le x \le\Theta,\\ 0 ...
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Average wait time arriving at subway randomly
If the subway comes every 10 minutes on average, what is the expected wait time if I arrive at the station randomly? Can someone help me mathematically understand this problem?
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Proof of upper-tail inequality for standard normal distribution
$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$,
$$
\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.
$$
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2answers
246 views
Name this paradox about most common first digits in numbers
I remember hearing about a paradox (not a real paradox, more of a surprising oddity) about frequency of the first digit in a random number being most likely 1, second most likely 2, etc. This was for ...
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3k views
How many rolls do I need to determine if my dice are fair?
Roughly how many times do I need to roll a 6-sided die to feel confident that it's giving "fair" results? What about a 10-sided or 20-sided die?
Note that I will be actually manually rolling physical ...
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2answers
163 views
Is there a name for the matrix $X(X^tX)^{-1}X^{t}$?
In my work, I have repeatedly stumbled across the matrix (with a generic matrix $X$ of dimensions $m\times n$ with $m>n$ given) $\Lambda=X(X^tX)^{-1}X^{t}$. It can be characterized by the ...
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4answers
762 views
How can I calculate “most popular” more accurately?
I'm developing a website at the moment.
The website allows users to "rate" a post from 0 to 5.
Posts can then be displayed in order of popularity.
At the moment, my method of calculation is pretty ...
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4answers
819 views
Geometric mean never exceeds arithmetic mean
This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)
The geometric ...
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2answers
130 views
Improper integral of $\frac{x}{e^{x}+1}$
The improper integral of $\frac{x}{e^x-1}$ (along the positive real line) comes up in a lot of places, you can even invoke the Riemann-zeta and Gamma functions to solve it nicely.
However, I just ...
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5answers
266 views
What does it mean to do MLE with a continuous variable
I am struggling with the semantics of continuous random variables.
For example, we do maximum likelihood estimation, in which we try to find the parameter $\theta$ which, for some observed data $D$, ...
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1answer
133 views
$X$, $Y$ gaussian variables, $\mathbb{E}[X^2Y]$ and $\mathbb{E}[X^3Y]$ as a function of its means, variances and covariance?
Let be X and Y two not independent Gaussian random variables of means $\mu_X$, $\mu_Y$ and variances $\sigma_X$, $\sigma_Y$, respectively. Let also be $\Sigma$ the covariance between X and Y.
I'd ...
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3answers
145 views
reason of the definition of the covariance
The covariance of two random variables $X$ and $Y$ is defined to be
$${\rm Cov}(X,Y) = E[(X-E[X])(Y-E[Y])]. $$
I don't understand it, if someone could explain me this please.
Why does this value ...
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3answers
257 views
Usefulness of Variance
I've had a look for intuitive explanations of the variance of an RV (e.g. Intuitive explanation of variance and moment in Probability.) but unfortunately for me, I still don't feel comfortable with ...
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1k views
How to accurately calculate erf(x) with a computer?
I am looking for an accurate algorithm to calculate the error function. I have tried using this formula (Handbook of Mathematical Functions, formula 7.1.26), but the results are not accurate enough ...
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129 views
About joint probability divided by the product of the probabilities
Let $X$ and $Y$ be two events.
So $P(X)$ is the probability of $X$ happens, and $P(Y)$ is the probability of $Y$ happens.
So $P(X,Y)$ is probability of both $X$ and $Y$ happen.
So what is the ...
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6answers
2k views
What does -1.13 times faster mean?
I'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books.
The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I ...
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4answers
16k views
How to calculate percentile? Is it possible to get 100 percentile?
How do we calculate percentile? I think it should be calculated as:-
P:- Total number of candidates
L:- Number of candidates whose marks are below yours
Percentile = (L/P)*100
That means if you are ...
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2answers
1k views
Characteristic function of the normal distribution
The standard normal distribution
$$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$$
has the characteristic function
$$\int_{-\infty}^\infty f(x) e^{itx} dx = e^{-\frac{t^2}{2}}$$
and this can be ...
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4answers
638 views
Question about basic strategy in Blackjack
I was watching Beating Blackjack with Andy Bloch where he runs through the basic strategy charts that outline the best strategy with playing the game. Later he also talks about the methodologies to ...
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4answers
337 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 ...
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1answer
144 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 ...
