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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How do you estimate the mean of a Poisson distribution from data?

I have thought of three different approaches for estimating the mean for a Poisson, but I am not sure which one is the correct method to estimate it (the third one is documented separately at the end ...
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921 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 ...
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How do I calculate the odds of a given set of dice results occurring before another given set?

Dice odds seem simple at first glance, but I've never taken a Calculus based statistics course or game theory, and I think I may need to in order to solve some of the things I'm trying to solve. I can ...
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Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
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Improbable vs Impossible?

I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossible ...
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607 views

Name of “De Moivre's equation”

This article refers to something it calls "De Moivre's equation": $$\sigma_{x} = \sigma / \sqrt{n}$$ Basically, it relates the observed variance in a sample size to the actual variance of the ...
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Finding the maximum likelihood estimator

Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty<x<\infty$$ What I ...
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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 ...
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Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
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What does multiplication mean in probability theory?

For independent events, the probability of both occurring is the product of the probabilities of the individual events: $Pr(A\; \text{and}\;B) = Pr(A \cap B)= Pr(A)\times Pr(B)$. Example: if you ...
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307 views

Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
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180 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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Why is the Kullback-Leibler divergence not symmetric?

As known the Kullback-Leibler Divergence: $$\operatorname{KL}=\sum_{i=1}^n \ln(\frac{P(i)}{Q(i)})P(i)$$ is not symmetric. I would like to know how this can be seen from the formula. I am aware that ...
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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. ...
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Example of Sufficient and Insufficient Statistic?

I am having trouble understanding the concept of a sufficient statistic. I have read What is a sufficient statistic? and Sufficient Statistic (Wikipedia) Can someone please give an example of: a ...
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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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Kendall tau calculation

Can someone explain how the Kendall tau works? I can't seem to find a good explaination/tutorial/example. I've been running corr(x,y,'kendall') from Matlab's ...
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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 $\rho \in [-\frac 12,1]$, ...
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481 views

Is there a way to check the correctness of your answer to a probability question?

In CS, there's a systematic way to check if your code is buggy or not as you write code. Is there a way to check the correctness of your answer to a probability question without using a textbook? For ...
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446 views

Function of a random variable: expectation

Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. random variables (i.e. a random sample) with pdf: $$f_X(x) = e^{-(x-\theta)} \, e^{-e^{-(x-\theta)}} · \mathbf{1}_{x\in \mathbf{R}}$$ The goal is ...
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749 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)] = ...
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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 ...
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Which theory is used to calculate the position and energy of a point source?

Consider an empty room with one point source that emits a stationary signal (constant sound, radioactive radiation, ...). The energy nor the position of the point source is known. We send someone in ...
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Chances of someone being of a certain gender at websites

I have 2 of websites and I know the chances of a visitor being a female or male. Let's say I have 2 website where the chance of a new visitor being a female is 80%. If the visitor comes on website 1 ...
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664 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 ...
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The expectation of absolute value of random variables

I need some help with the following problem: Let $X_1,...,X_n$ be a random sample from Normal$(0,1)$ population. Define $$Y_1=| {{1 \over n}\sum_{i=1}^{n}X_i}|, \ Y_2={1 \over ...
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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 ...
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344 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 ...
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281 views

What is a confidence interval?

What are the nature and purpose of confidence intervals?
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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 ...
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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. ...
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Given every horse's chance of winning a race, what is the probability that a specific horse will finish in nth place?

I have been interested in calculating a specific horse's chance of finishing in nth place given every horse's chance of winning in a particular race. i.e. Given the following: ...
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sum of two independent exponential distribution

let $Y_1\sim \exp(\lambda_1)$ and $Y_2\sim \exp(\lambda_2)$ and $V=Y_1+Y_2$ Show that the pdf of $p_V(x)$ of $V$ has the following form ...
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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 ...
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Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
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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 ...
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103 views

$L^1$ convergence of PDFs vs $L^2$ convergence of CDFs

Let $f_n$ denote a sequence of PDFs, and $F_n$ denote the corresponding sequence of CDFs. Given $L^1$ convergence of the PDFs to some PDF $f$, $$\int_\mathbb{R} |f_n(x) -f(x)| dx \rightarrow 0$$ ...
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Are there order statistics for a Gaussian variable raised to a power?

Let $X$ be a random variable with a standard normal distribution. Let $Y = |X|^{2p}$. I am trying to find the distribution for $Y_{(n)}$, i.e., the largest value of $Y$ out of $n$ samples. I have ...
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Concentration inequality for sum of squares of i.i.d. sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
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Linear regression: degrees of freedom of SST, SSR, and RSS

I'm trying to understand the concept of degrees of freedom in the specific case of the three quantities involved in a linear regression solution, i.e. $SST=SSR+SSE, $ i.e. Total sum of squares = ...
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262 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 ...
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Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why $R^{2}$ in the regression of $Y$ on $X =$ the sample correlation between X and Y the ...
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276 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 ...
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Check answer, How to find Cov(x,y) and Var(2x-y)?

I have the following tableau ...
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Limit of median of uniform distribution

Let $X_1,X_2,\ldots$ be a random sample from the uniform distribution on the interval $(0,1)$. Assuming that $n$ is odd, find the pdf of the sample median (say $M_n$). Does the pdf of the r.v. ...
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Spinners from yesteryear: A challenging probability problem

While browsing the Internet I found an old horse racing game where the results were determined by a spinner. The names of 6 different horses were listed an equal number of times on the spinner. Each ...
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139 views

Asymptotic efficiency of maximum likelihood estimate

Let us consider a simple statistical model $\{f_{\theta}\}$ where $\theta\in U$, an open subset of $\mathbb{R}$. Let $X_1,\dots,X_n$ be sample drawn from $f_{\theta}$. I know, under some regularity ...
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Intuition of Gamma Family

The function $$f(t) = \frac{t^{\alpha-1}e^{-t}}{\Gamma(\alpha)}, \ \ 0 < t < \infty$$ is a pdf. But Why is the gamma family defined as $$f(x| \alpha, \beta) = \frac{1}{\Gamma(\alpha) ...
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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 ...