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

9
votes
2answers
418 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
9
votes
1answer
371 views

On the probability of obtaining 27 sets in the card game Set

For people not familiar with the card game Set, see its entry on Wikipedia and/or one of the related questions here on Math SE. It might be faster to just play the game a couple of times though, see e....
9
votes
0answers
889 views

Maximum likelihood and sufficient statistics

$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$ where our mean is $C+B$ and $t>0$. so far i have found my log likelihood functions and differentiated them as follows: $$dl/dB = \sum[t\exp(t/C)...
8
votes
4answers
6k 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 ...
8
votes
2answers
12k views

Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
8
votes
4answers
3k views

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 ...
8
votes
4answers
11k 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 ...
8
votes
4answers
474 views

Probability with unknown variables

An urn contains $10$ red marbles and $10$ black marbles while a second urn contains $25$ red marbles and an unknown number of black marbles. A random marble will be selected from each urn and the ...
8
votes
4answers
14k views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
8
votes
3answers
290 views

How come in statistics there is very little justification for the formulas used and proofs are almost nonexistent [closed]

I don't understand why people accept certain formulas in statistics without a mathematical proof style argument. You see this a lot in statistics textbooks and unfortunately this spills over with the ...
8
votes
5answers
2k 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 ...
8
votes
4answers
6k views

How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula 7.1....
8
votes
3answers
699 views

Probability of getting a job when applying for 3 places

I have the following problem. You apply for jobs and know that if you send your application then every job appointment procedure has two stages: You can be either invited or not invited ...
8
votes
4answers
343 views

A statistical approach to the prisoners problem

Two days ago, I found this problem on reddit (I didn't have access to reddit when I did the math, so I did it with 24 instead of 23, and I decided the warden picked someone every day, not "whenever he ...
8
votes
1answer
15k views

density of sum of two uniform random variables $[0,1]$

I am trying to understand an example from my textbook. Let's say $Z = X + Y$, where $X$ and $Y$ are uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z & \text{...
8
votes
2answers
7k views

Proof of the independence of the sample mean and sample variance

I've been trying to establish that the sample mean and the sample variance are independent. One motivation is to try and write the sample variance, $S^{2}$ as a function of $\left\{ X_{2}-\bar{X},X_{3}...
8
votes
3answers
261 views

distribution of $X^2 + Y^2$

Suppose $X$ and $Y$ are independent uniform distributions between $(0,1)$. What is the distribution of $X^2 + Y^2$? I derived that the pdf of $X^2$ is $\frac{1}{2\sqrt{x}}$ for $0\leq x \leq 1$. How ...
8
votes
2answers
7k views

Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
8
votes
1answer
8k views

Proof that the hypergeometric distribution with large $N$ approaches the binomial distribution.

I have this problem on a textbook that doesn't have a solution. It is: Let $$f(x)=\frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}\;,$$ and keep $p=\dfrac{r}{N}$ fixed. Prove that $$\lim_{N \...
8
votes
2answers
86 views

Why does probability need the Axiom of pairwise disjoint events? [duplicate]

I'm a beginning student of Probability and Statistics and I've been reading the book Elementary Probability for Applications by Rick Durret. In this book, he outlines the 4 Axioms of Probability. ...
8
votes
3answers
477 views

Stein's lemma condition

(Apologies if I break some conventions, this is my first time posting!) I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...
8
votes
4answers
2k views

Estimate probabilities from its moments

I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its ...
8
votes
3answers
10k views

Is There Something Called a Weighted Median?

I was given some data that represents the number of lines in a document as well as the line count per hour (which is the lines in the document divided by the number of hours that the document was ...
8
votes
6answers
10k views

Great Book on Probability and Statistics (for Computer Scientists)

I'm a Computer Science sophomore and we're studying Probability and Statistics (fundamentals and all). The teacher recommends a book which I don't like since it does not even try and explain ...
8
votes
1answer
623 views

Monotonic behavior of a function

I have the following problem related to a statistics question: Prove that the function defined for $x\ge 1, y\ge 1$, $$f(x,y)=\frac{\Gamma\left(\frac{x+y}{2}\right)(x/y)^{x/2}}{\Gamma(x/2)\Gamma(y/2)}...
8
votes
1answer
177 views

relative size of most factors of semiprimes, close?

when chatting about RSA a cohort just asserted something like "most prime factors of semiprimes are roughly the same size" measured in bits. ie "bits" is the number of digits in the base2 ...
8
votes
1answer
397 views

What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
8
votes
1answer
16k 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 ...
8
votes
1answer
138 views

Heavy-tailed distributions

I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$: (i) $\limsup_{x\to\infty}\frac{\...
8
votes
2answers
169 views

Lies, damned lies, and statistics

A story currently in the U.S. news is that an organization has (in)conveniently had several specific hard disk drives fail within the same short period of time. The question is what is the likelihood ...
8
votes
3answers
547 views

Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
8
votes
1answer
161 views

Expected value and variance of ratio of two sums of two sets of random variables

Let $X_1,X_2,\ldots,X_n$ be iid $\operatorname{Gamma}(\alpha,\beta)$ random variables. Suppose that, conditionally on $X_1,X_2,\ldots,X_n$, the random variables $Y_1,Y_2,\ldots,Y_n$ are independent ...
8
votes
1answer
160 views

If half the population were murderers, and they could only kill once, how many would survive?

So here's the rules: Half the population are murderers Each murderer can only kill once We assume the nobody will fight back, and only murderers can murder Murderers can kill other murderers Only ...
8
votes
2answers
602 views

Can the maximum likelihood estimator be unbiased and fail to achieve Cramer-Rao lower bound?

If some maximum likelihood estimator (MLE) turns out to be unbiased (which does not necessarily holds), then does it achieve the Cramer-Rao lower bound (CRLB) even in finite sample? (It does when the ...
8
votes
0answers
670 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
7
votes
5answers
3k views

Arithmetic mean. Why does it work?

I've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the number ...
7
votes
2answers
1k 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 &...
7
votes
2answers
245 views

Finding $E\left[\frac{\sum_{i=1}^n X_i^2}{(\sum_{i=1}^n X_i)^2}\right]$ of a sample of gamma random variables

Suppose $X_1,\ldots,X_n$ is a random sample from the $\Gamma(k,\lambda)$ distribution where $\lambda$ is unknown and $k$ is a positive integer and known. How can I find $$E\left[\frac{\sum_{i=1}^n X_i^...
7
votes
3answers
1k views

What is the expected number of dice one needs to roll to get any monotonically increasing series of 1 to 6?

Similar to: "What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?" but we allow repeats so 1,1,2,2,3,4,4,4,4,5,5,6 would count. My answer (or simulation) is flawed as I ...
7
votes
4answers
7k views

What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
7
votes
5answers
2k views

Coin tosses until I'm out of money

The question I think is a simple one, but I've been unable to answer or find an answer for it yet: There's a simple game: if you flip heads you win a dollar (from the house), but if you flip tails ...
7
votes
1answer
20k views

What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
7
votes
1answer
236 views

Is pi lying on the ground, and on TV?

Consider the leaves from a bunch of trees in a terraced plaza in the Autumn. It may well happen that the tiles of the terrace are squares whose length easily exceeds the length of the stem of the ...
7
votes
2answers
2k views

What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
7
votes
5answers
15k views

Given a data set, how do you do a sinusoidal regression on paper? What are the equations, algorithms?

Most regressions are easy. Trivial once you know how to do it. Most of them involve substitutions which transform the data into a linear regression. But I have yet to figure out how to do a ...
7
votes
2answers
5k views

Prove that the sample median is an unbiased estimator

My book says that sample median of a normal distribution is an unbiased estimator of its mean, by virtue of the symmetry of normal distribution. Please advice how can this be proved.
7
votes
2answers
5k views

Finding expected number of distinct values selected from a set of integers

I have a set of $n$ integers $\{1, . . . , n\}$, and I select three values with replacement. How can I find the expected number of distinct values? Note each value is chosen uniformly and ...
7
votes
2answers
5k views

Calculate variance from a stream of sample values

I'd like to calculate a standard deviation for a very large (but known) number of sample values, with the highest accuracy possible. The number of samples is larger than can be efficiently stored in ...
7
votes
2answers
682 views

The birthday paradox [duplicate]

I would like a better understanding of the famous birthday paradox. "What is the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday?" I understood ...
7
votes
5answers
1k views

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 ...