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

0
votes
0answers
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 ...
2
votes
3answers
929 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 ...
1
vote
3answers
262 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 ...
0
votes
3answers
664 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 ...
2
votes
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. ...
0
votes
1answer
247 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 ...
0
votes
2answers
515 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 ...
2
votes
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 ...
1
vote
1answer
812 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 ...
0
votes
1answer
78 views

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 ...
0
votes
1answer
248 views

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 ...
1
vote
2answers
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 - ...
1
vote
2answers
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 ...
1
vote
2answers
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 ...
4
votes
1answer
122 views

Calculation of an expectation for the 'part' of a vector

Let $x$ be vector in $R^n$. Let $\pi(⋅)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Calculate $$ E\left|\sum_{i=1}^mx_{\pi(i)}\right|^q, ...
1
vote
2answers
84 views

Finding the distribution of an equation

I am new to this site so I'm not quite sure how this whole thing goes. Anyway, I have a question from my homework for statistics class. I'm new to statistics so questions that seem easy to you might ...
3
votes
3answers
1k views

Probability question (arrangements)

Question: A child has 12 blocks. 6 are black, 4 are red, 1 is white, and 1 is yellow (A) If the child puts the blocks in a line, how many different arrangements are possible? (B) If one of the ...
9
votes
4answers
6k views

How would I figure out how many anagrams of mississippi don't contain the word psi?

I'm really confused how I'd calculate this. I know it's the number of permutations of mississippi minus the number of permutations that contain psi, but considering there's repetitions of those ...
1
vote
2answers
66 views

Variance decomposition over pairs of elements

how can I prove that (if it is correct) ? $\sum_{u,v \in P \times P} \frac{|u-v|^2}{N^2} = 2 \cdot Var(P)$ where $N$ is the number of elements of $P$. $P$ is a list of numbers.
2
votes
1answer
123 views

A question about Poisson and Binomial distributions

I'm struggling with understanding why the following statement is true: Let $X$ be a Random Variable with Poisson distribution. Let $Z$ be a Random Variable independent from $X$, whose distribution ...
0
votes
3answers
882 views

Integrating the pdf of a normal distribution

I need to find the distribution of $Y=X_1+X_2$ where both $X_1$ and $X_2$ are normally distributed with $(\mu,\sigma^2)$. So I'm looking for ...
0
votes
1answer
62 views

What does $P^3(D,D)$ stand for?

We're on Markov and we're considering the Markov chain on $\{A,B,C,D,E\}$ with a transition matrix $P$. I am asked to find $P^3(D,D)$ but I am unfamiliar with what the notation stands for. Thanks
1
vote
0answers
37 views

Verify correct use of Chi-Squared method

I have a data set which I obtained from experiment. I which to compare this with expected values obtained from $y=2^x$. Could this be done with the Chi-squared distrobtion: $$\begin{array}{c|c|c} ...
1
vote
2answers
83 views

Probability generating function $5$

In a game of chance one of the numbers $1,2,3$ appears. The game is played five times and the total score is recorded. If the probs of $1,2,3$ are $\frac{1}{6}, \frac{1}{3}$ and $\frac{1}{2}$ ...
0
votes
2answers
52 views

Testing goodness of fit of data and function

I have a data set which I obtained from experiment e.g. $$\begin{array}{c|c|c} \text{x} & \text{y} \\ \hline \\1 & 1.01 \\2 & 3.99 \\3 & 8.86 \\4 & 16.02 \end{array}$$ etc. ...
4
votes
2answers
1k views

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 ...
0
votes
1answer
2k views

Creating Venn Diagram To Aid In Solving Probability Question.

The question is: Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad non-wetting, ...
1
vote
0answers
45 views

maximising the frequency of mode.

I have 4 numbers 5,5,3,1. Now I have the number 5, which I can distribute in any manner to ...
0
votes
2answers
202 views

Probability generator

$A$ and $B$ play a game in which they take it in turns to throw a quoit over a peg. At each throw $A$ has a probability of $1/3$ of succeeding, whilst $B$ has a probability of $1/4$ of succeeding. ...
0
votes
1answer
36 views

Poisson distribution 3

Two shops $A$ and $B$ have accidents occurring randomly at a rate of two accidents in three months in $A$ and three accidents in four months in $B$. I come to shop $B$ on 11 Jan; I have no effect ...
0
votes
1answer
1k views

Probability of Purchasing

The question is, "Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store. (a).If the probability that at most one of these purchases an ...
0
votes
1answer
253 views

Weak Law of Large Numbers

The given problem is as follows: Recall that $\log 2 = \int_0^1 1/(x+1) dx$. Hence, by using a uniform $(0,1)$ generator, approximate $\log 2$. Obtain an error of estimation in terms of a large ...
2
votes
1answer
2k views

How do I find the MLE of $\theta$ when x is dependent on $\theta$?

Let $X_{1},X_{2},...,X_{n}$ represent a random sample from a distribution with pdf: $f(x; \theta)=e^{-(x-\theta)}, \theta \le x<\infty, -\infty<\theta<\infty$ | zero elsewhere I need to ...
0
votes
1answer
910 views

median of a uniform distribution [0,1]

I need to find the distribution of the median from the given distribution, where n is known to be odd. The formula given in class for this is: $n=2m+1$ where $m\in\mathbb{N}$ ...
0
votes
1answer
138 views

Does Principal Component Regression still work in high-dimensional ($N<p$) situation?

I understand that, many classical methods for multiple regression won't work when $N<p$, where $p$ is the dimension of the input space and $N$ is the sample size. For example, LSE for multiple ...
0
votes
1answer
3k views

Method of Moments on a Uniform distribution (a,b)

Given that $x_1,x_2,...,x_n$ are i.i.d. random variables drawn from a Uniform distribution over $(a,b)$ (with $a<b$ and both are unknown), I hope to estimate $a$ and $b$ using the method of ...
2
votes
1answer
246 views

Calculating the 4th number from 3 related numbers

I was taught a long time ago this really useful trick to calculate an unknown value when you know 3 existing values that are related. An example would be something like: If there are 21 peas in 3 ...
2
votes
1answer
2k views

Finding the Moment Generating Function for a random variable X^2

If X~(0,1), integrate to find the moment generating function of a random variable $X^2$ and identify the distribution of $X^2$ using the moment generating function. ...
0
votes
1answer
300 views

moment generating functions by integration

Let X~N(0,1)m find the moment generating function of $X^2$ using integration techniques. I'm not sure exactly what this is asking me to do. Is $X^2$ just the pdf for the standard normal function ...
1
vote
3answers
2k views

Advantage of Relative Standard Deviation (RSD)

The definition of RSD is given below: Relative Standard Deviation: In probability theory and statistics, the relative standard deviation (RSD or %RSD) is the absolute value of the coefficient ...
2
votes
2answers
625 views

Geometric Distribution $P(X\ge Y)$

I need to show that if $X$ and $Y$ are idd and geometrically distributed that the $P(X\ge Y)$ is $1\over{2-p}$. the joint pmf is $f_{xy}(xy)=p^2(1-p)^{x+y}$, and I think the only way to do this is to ...
0
votes
1answer
50 views

$u$~$N(0,A)$ and z$|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$?

$u$~$N(0,A)$ and $z|u$~$N(u,1)$ how to show that $u|z$~$N(Bz,B)$ where $B=A/(A+1)$ ?
0
votes
1answer
48 views

measure distance between peaks in a pdf

I'm looking for a measure/score for the following case: I have a 1D vector of size s. The vector has values (probabilities) in several places, whilst most of the vector is zeros. I need some kind of ...
5
votes
5answers
2k 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 ...
2
votes
1answer
49 views

What are the theorems on the inevitability of some kind of order in large sets?

I've read Paulos' A Mathematician Plays the Stock Market: The problem is that if you look hard enough, you will always find some seemingly effective rule that resulted in large gains over ...
3
votes
1answer
98 views
1
vote
2answers
60 views

How would I determine the number of solutions for the sum of this number?

We're given the following equation: $$x_1 + x_2 + x_3 + x_4 + x_5 = 20$$ I know that the simple amount of solutions to this is $\binom{5+20-1}{5-1}$, but for the following questions I'm slightly ...
3
votes
3answers
82 views

I'm asked to give a combinatorial proof of the following question, how would I go about doing it?

The hint we're given is to think of it in terms of grid paths. I don't necessarily want the answer, I'm interested in learning how to do this. Could anyone offer an explanation (clearly please) on ...
2
votes
1answer
106 views

How would I go about solving this statistics question on permutations?

Columba has two dozen each of n different coloured beads. If she can select 20 beads (with repetition of colors allowed) in 230,230 ways, what is the value of n? I'm trying to figure it out, ...
1
vote
1answer
201 views

A problem about UMVUE

I got a problem as follows: If $\overline X_1$ is the mean of a random sample of size $n$ from a normal distribution with the mean $\mu$ and the variance $\sigma_{1}^2$, $\overline X_2$ is the ...