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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Finding the efficiency of an unbiased estimator

I have a random sample drawn from a $N(\theta,\sigma^2)$ distribution with $\sigma^2$ known. I am trying to estimate $\theta$. I need to calculate the efficiency of the unbiased estimator, ...
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115 views

Construct a Confidence Interval of $95\%$

Based on a random sample of 20 values from a normal distribution with mean $\mu$ and variance $\sigma^{2}$, it was calculated that $\bar{X}=8$ and $s=4$. Provide a $95\%$ confidence interval for the ...
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Does this count as a Monte Carlo simulation?

Let's say I have a group of robots that walk on a 11x11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different ...
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247 views

Game Theory: determining the value of the foxhole game

"A soldier can hide in one of five foxholes, and a gunner can hide in four spots: A, B C, and D. The configuration looks like this: 1 (A) 2 (B) 3 (C) 4 (D) 5. If a shot is fired at a location and the ...
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473 views

On the empirical mean and variance of a Poisson i.i.d. sample

Let $X_1, X_2, \ldots, X_n$ be a random sample from a Poisson($\lambda$) distribution. Let ($\bar{X}$) be their sample mean and $s^2$ their sample variance. Show that ...
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126 views

How can I solve this integral?

How can I solve the following integral? $$\int_{-\infty}^\infty \prod_{i=1}^n \bigg( 1 - \Phi\left(\frac{c - \mu_i}{\sigma_i}\right) \bigg) \frac{1}{\sigma_Y}\phi \bigg(\frac{c-\mu_Y}{\sigma_Y} ...
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491 views

Conditional Probability Problem With Non-Disjoint Events

The problem is: A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let Bbe the event that the ...
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1k views

Conditional Probability Problem With Boxes

The problem I am working on is: One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the first box ...
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58 views

Gambler tosses coin PGF

A gambler tosses a coin repeatedly until it comes up tails. He gets £1 for each head that comes up before that happens, and pays back £1 for the tail. Find the probability generating function for his ...
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663 views

Probability of 5 Card Poker Hand with Only Two Suits (Method)

Probability of 5 Card Poker Hand with Only Two Suits: Pick a card from any suit: 52 Pick a card from a different suit: 39 Pick three more cards from either of the first two suits: $24*23*22$ So, ...
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549 views

Intuition Of Conditional Probability Equation

I was wondering if any one of you had any intuitive insight regarding the conditional probability equation, $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$. In my textbook, they give a mere definition, ...
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531 views

How to count $n$th percentile from normally distributed random variable?

I have normally distributed random variable $X\sim \mathcal N(100,225)$. How to count $n$th percentile? In my case I need lower quartile - $x(0.25)$.
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Combinatorics and Probability Problem Concerning Poker Hands

The problem I am currently working on is: In five-card poker, a straight consists of five cards with adja-cent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king ...
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1answer
199 views

Given the sample that not Independently and Identically Distributed

So I'm working on a question that wants me to consider the sample mean $\mu_1$ (estimator of $\mu$) when $n=3$ to be $μ_1=X_1+X_2+X_3$. Now first I assume that they are independently and identically ...
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7k views

How to check if my dataset is normally distributed?

I have data sets (measurements) and I need to know if values are normally distributed. I would like to get this information programmatically in my application and not via plotting and checking it ...
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1answer
18 views

Clarification on a basic stat problem

I need a clarification on this homework problem I'm getting done. The question is as follows: For each positive integer $n$, let $P_n = \Big (\frac 12 \Big)^n.$ Consider the events $A = \{n:1 \leq ...
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103 views

How to estimate parameters of a normal distribution?

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 ...
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243 views

Probability that the third student didn't answer correctly given that only 2 out of 3 students answered correctly

I tried solving this but failed many times. I couldn't really wrap my head around the right point to start. Here's the question: Students A, B, and C each independently answer a question on a test. ...
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290 views

Normal Distribution Identity

I have the following problem. I am reading the paper which uses this identity for a proof, but I can't see why or how to prove its true. Can you help me? \begin{align} \int_{x_{0}}^{\infty} e^{tx} ...
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1k views

Probability of repeating at least 2 digits

Pin codes consist of 4 digits between 0 and 9. If a pin-code were generated by a random number generator (e.g. by 4 ten-sided dice), what is the probability that it will have at least two digits that ...
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161 views

Graph of pdf and calculate moments

http://www.dumpt.com/img/viewer.php?file=0pt0l719ug05g1r09sly.jpg The diagram in the link shows the graph of a probability density function. Given that the mean is zero, express $b$ in terms ...
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Huffman minimum variance coding

it is well known that Huffman code with minimum variance is preferable. I've digged through entire Polish/English internet and this is what I found: to build Huffman code with minimum variance you ...
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2k views

Intersection of three events

I am working on a probability problem involving three events, A B and C, I am given Pr(A), Pr(B), Pr(C), Pr(A), Pr(A $\bigcup$ B), Pr(A $\bigcup$ C),Pr(B $\bigcup$ C) and we are asked to find ...
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Combinatorics and Probability Problem

The problem I am working on is: An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession. a.How many different possible PINs are there if ...
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120 views

Order Statistics : What's the deal with $n!$?

Assuming I have $n$ IID Random Variables $X_1,...,X_n$ the Order Statistics of this set $:= (Y_1,...Y_n)$ has a distribution of $f_Y(y_1,...,y_n)=n!f_X(x_1,...,x_n)$ I have read a couple of ...
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822 views

Probability of A Specific Type of Experiment Occuring

The problem is: An experimenter is studying the effects of temperature, pres-sure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different ...
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763 views

Central limit theorem to approximate a Poisson distribution.

Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in $1957$. Included as part of the test were some $3000$ military and civilian ...
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439 views

Average percentage of percentage

I ran a survey on my website where users marked a checkbox for what % they did X. Something like this. ...
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52 views

Box Plot Log Scaled?

When making box plots to represent data, is it logical to set our axis scale to logarithmic? Or does that defeat the whole point of a box plot?
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259 views

Finding the limiting distribution

I was given an exercise to do that sounded something like this: The Arizona football team scored $45$ goals in $19$ games in the 2007/08 season. If $y_i$ denotes the number of goals scored in the ...
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2answers
341 views

Probability distribution functions: factorization 3-way implies 2-way?

I recently asked a question about pairwise versus mutual independence (also related to this and this q). However, (1) I inadvertently used incorrect terminology: three events, A, B, C are ...
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1answer
3k views

Combinatorics Problem: To order or not to order.

The question I am working on is: A Little League team that has 15 players on its roster. a.How many ways are there to select 9 players for the starting lineup? b.How many ways are there to select 9 ...
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58 views

I want to find the real number c for which we have: $P(X<c)=P(Y<c)$

Consider two normal random variables $X$ and $Y$: $$X\sim N(m_1,s_1), \qquad Y\sim N(m_2,s_2)$$ I want to find the real number $c$ for which we have: $$P(X<c)=P(Y<c)$$
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Combinatorics Problem with Number of Web Sites

The problem I am working on is: As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com). a. How many domain names consisting of just two letters in sequence can ...
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1answer
230 views

A question about the convergence of distribution

Let $\bar{X_n}$ denote the mean of a random sample of size n from a distribution that has pdf $f(x) = e^{-x}$, $0<x<\infty$, zero elsewhere. a) Show that the mgf of $Y_n=\sqrt{n}(\bar{X_n}-1)$ ...
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1answer
661 views

Finding the cumulative distribution function for the double exponential

Im considering the double exponential with parameter $\lambda$, $$g(x)=\frac{\lambda}{2}e^{\lambda x}, x<0; \frac{\lambda}{2}e^{-\lambda x}, \geq 0$$ Just simple one looking for the c.d.f, I ...
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48 views

Quartiles - what computation method is correct?

I am not sure what method use for computation a lower and upper quartil. In my textbook there is a rule for lower quartil -"lower quartil is median of lower half of data" and "upper quartil is median ...
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874 views

estimate population percentage within an interval, given a small sample

Given a small sample from a normally-distributed population, how do I calculate the confidence that a specified percentage of the population is within some bounds [A,B]? To make it concrete, if I ...
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3answers
255 views

Simple standard deviation question using stocks as example

The following table is from page 171 of Fundamentals of Investing (11th edition) by Gitman, Joehnk, Smart. Please consider only the X, Y and XY columns (second, third, fifth). Portfolio XY comprises ...
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641 views

Combinatorics/Probability Example

I am working through the example problem the author of my statistics book provides: A particular iPod playlist contains 100 songs, 10 of which are by the Beatles. Suppose the shuffle feature is used ...
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4answers
1k views

Combinatorics Statistics Question

The problem I am working on is: An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee. ...
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1answer
234 views

Probability of having at least one pair in a baccarat game? [closed]

8 decks of playing cards are used in baccarat initially. In each game 4 cards are drawn from the shoe which contains the 8 decks of cards (416 cards in total), in which 2 cards for Banker and 2 cards ...
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502 views

Help to understand false positive ratio

From wikipedia: In statistics, when performing multiple comparisons, the term false positive ratio, also known as the false alarm ratio, usually refers to the probability of falsely rejecting ...
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1answer
53 views

Prove $\sum_{i=1}^n$$w_i^2\geq\frac{1}{n}$ given $\sum_{i=1}^n w_i=1$

I was looking at my stats textbook and they claim that the sample variance of a weighted distribution involving i.i.d. $x_i$s will be smallest when each of the weights is equal. I follow this argument ...
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770 views

Generating Sets From Given Information

The problem I am working on is: The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request ...
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Combining uniform (?) distributions

$C$ and $D$ make widgets. $C$ makes widgets with expected mass 10 kg and a standard deviation of 1kg and he makes 7 in an hour. $D$ makes widgets with expected mass 12 kg and a standard deviation ...
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How can you show this is a proper distribution ??

I derived a posterior distribution, which looks like this: $$p(\alpha, \beta | y,t) \propto \frac{exp( - \sum_i (\alpha + \beta t_i))\prod_i (\alpha + \beta t_i)^{y_i}}{\prod_i y_i!}$$ How can I ...
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91 views

Skewness - Zero

Why is the skewness of the samples $[ 1,\,2,\,3,\,4,\, 5,\, 6,\, 7,\, 8 ,\, 9 ]$ zero? Of course, it follows from definition, $\displaystyle E \left[\left(\frac{x_i - ...
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81 views

Probability generating function $6$

A symmetrical spinning top has five edges number -2, -1, 0, 1, 2. The PGF for the scores when the top is spun once is written $\frac{1-t^5}{5t^2(1-t)}$. Hence find the probability of getting a ...
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66 views

Why is this statement true in probability/statistics?

$P(X_n \leq X, |X_n - X|< \epsilon) \leq P(X \leq X + \epsilon)$ because $X_n - \epsilon < X < X_n + \epsilon < X + \epsilon$ For the right side of the equality...since we are not ...