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

-1
votes
1answer
30 views

Second moment of a PGF and relation to expectation

Given two random discrete variables X and Y and their probability generating function Q(x,y), I need to write the following expression in terms of E[X] and E[Y]: $$\ \frac{dQ(x,y)}{dxdy^2} = \ ...
1
vote
1answer
21 views

Continuous Percentage

This is probably super easy for the experts here (I need to retake and practice a lot of math courses, its been a while). Suppose I have a shared bank account with N number of people. Anyone can add ...
1
vote
1answer
25 views

Does Variance depend on change of scale and origin?

I know that SD depends on change of scale and not on change of origin. But what about Variance? And why?
0
votes
0answers
35 views

Statistics Question. Method of Maximum Likelihood, Finding MLE's

This is my first ever question. Details on what/why you are doing what you are doing would be amazing. I saw a question with the same problem but did not understand the answer NOR was part (b) asked ...
0
votes
0answers
13 views

If $X_1,X_2$ are iid U($\theta$,$\theta+1$) find $P(X_1+X_2 > C)$

I need to find the probability, $P(X_1+X_2 > C)$ Where $X_1$ and $X_2$ are U($\theta$,$\theta+1$). The solution I have found gives the following, $\int_{1-C}^{1}\int_{C-x_1}^{1} 1 dx_1 dx_2$ ...
2
votes
1answer
69 views

Simple calculations of mean, standard deviation, and probability

You are a successful entrepreneur that has developed a new sustainable product that is manufactured through a standard production process. As part of this process, the product goes through quality ...
0
votes
1answer
65 views

Let X = the time between two successive arrivals at the drive-up window of a local bank…

Let $X$ = the time between two successive arrivals at the drive-up window of a local bank. $X$ has an exponential distribution with $\lambda = 2$. That is the probability density of $X$ is $f(X | ...
3
votes
2answers
31 views

Intuitive explanation for dividing by n-1 when calculating sample variance?

I understand how to mathematically show that the sample variance (that involves dividing by n-1) is an unbiased estimator of the population variance (which divides by n), and the mathematics has been ...
0
votes
0answers
5 views

Usage of error bars to denote confidence - when to use postive, negative, or both?

I was hoping someone would be able to give me a brief answer on when to use different types of error bars in a graph. I've seen three different types: error bars that only go in the negative y-dir, ...
0
votes
2answers
28 views

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve

Let Z be a standard normal random variable and calculate the following probabilities, indicating the regions under the standard normal curve? a) $P(0 < Z < 2.17)$ b) $P(-2.50 < Z < ...
1
vote
0answers
22 views

Randomly distributing numbers on a line, controlling density

This problem is for me quite difficult to describe - and hence I can't easily google it, so forgive me for not being able to google. Now it is easy to create a random distribution, say we have a ...
0
votes
1answer
25 views

Consider a communication source that transmits packets containing digitized speech…

Consider a communication source that transmits packets containing digitized speech. After each transmission, the receiver sends a message indicating whether the transmission was successful or ...
2
votes
0answers
84 views

Properties of eigenvectors of a sample covariance matrix?

My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the ...
1
vote
1answer
40 views

Why the principal components correspond to the eigenvalues?

Suppose ${\bf{X}} = ({X_1},{X_2},\ldots,{X_n})$ are the original components (also random variables) and ${{\bf{w}}_j} = ({\omega _1},{\omega _2},\ldots,{\omega _n})$ are loadings for the $j$th ...
0
votes
2answers
13 views

If X~beta($\theta$,1) then Z=-ln(x) ~exp(1/$\theta$)

I keep getting this, $P(Z\leq z)=P(-ln(X) \leq z) = P(X \leq e^{-z}) = \int_{0}^{e^{-z}} \theta t^{\theta-1}dt = t^{\theta}|_{0}^{e^{-z}}=e^{-z\theta}$ $\frac{d}{dz}[e^{-z\theta}]=-\theta ...
0
votes
0answers
23 views

Chi Square Formula and Degrees of Freedom Questions

I have a population sample with 200 points of data and 3 degrees of freedom so am I supposed to do a chi square formula with all 200 points of data? I believe that is what I'm supposed to do but I'm ...
2
votes
1answer
37 views

Is it possible to have two lines of best fit?

Could you rig a data set to have two lines of equally good (and best) fit? Or is it impossible?
0
votes
1answer
16 views

What is the probability of arriving at a traffic light when it is red if the red signal flashes for 30 seconds, the yellow

What is the probability of arriving at a traffic light when it is red if the red signal flashes for 30 seconds, the yellow signal for 5 seconds, and the green signal for 45 seconds Let red be R , ...
-1
votes
0answers
11 views

How to calculate main effects for a full factorial design with 5 factors?

I have data that has 5 factors, each 2 levels. The data is the same as what is on this page. http://www.itl.nist.gov/div898/handbook/pri/section4/gifs/ceramic.txt How am I to go about calculating the ...
3
votes
0answers
33 views

showing that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$. [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
3
votes
2answers
121 views

Correlation of Proportions

To introduce my question, here is a small simplification for consideration: Let $X,Y$ be independent random variates, each with finite mean and variance. Interestingly, ...
1
vote
0answers
30 views

Negative Mahalanobis distance

I am using Mahalanobis distance to compute a observation say: $x$ to a bunch of observations say: $\{x_1, x_2, ..., x_k\}$for classification. $\{x_1, x_2, ..., x_k\}$ has mean $m$ and covariance ...
0
votes
0answers
15 views

Is there any theorem on the convergence of area integral and L2 inner product

Suppose that $f$ is an unknown function on $R^d$ and $\hat f(x)=\sum_{i=1}^N w_i\delta(x-x_i)$ is a discrete approximation of $f$ given by some estimation method, where $\delta(x)$ denotes the Dirac ...
0
votes
1answer
16 views

Statistical test to evaluate if sample standard deviation differs from population standard deviation?

Beginner in stats. I have read about tests that allow you to estimate with a certain degree of confidence whether a sample mean is consistent with a population mean. But I'd like to know is there an ...
0
votes
0answers
8 views

Distribution of $Mx$ where $M$ is a Haar matrix and $x$ is a unit vector

I have a $n\times N$ random matrix $R$ with entries i.i.d. standard Gaussian. $n<N$. Then the matrix $M=(RR^T)^{-1/2}R$ is Haar distributed, i.e. has orthonormal rows and uniformly random ...
0
votes
1answer
40 views

Generate a uniform distribution from n coin flips

I'm making a computer game and I've reduced the problem into something simple: How can I show the player the number of heads he "tossed" given some number of coins = n? Naive expected value is ...
1
vote
1answer
36 views

How many cycles $A$ and $B$ can form this cycle

How many cycles $A$ and $B$ can form this cycle: $AB=(axyguimjrcwk)(bvqphsleofzt)(d)(n)$ I can see that $A$ and $B$ must share the cycle $(dn)$, and I believe due to ordering, both $A$ and $B$ must ...
0
votes
2answers
29 views

Probability one proportion is greater than another?

Candy maker makes child, and adult bags of jelly beans with different color mixes. The company claims, the child size is 30% red, and the adult size is 15% red. Assume their claim is true. Suppose we ...
1
vote
0answers
10 views

How to determine values of coefficients for a comparisonx using factorial design?

I think my problem is best answered by answering the following example but the more general the answer and explanation the better: Given 2 factors X and Y, x with 2 levels x1 x2 and Y with three ...
0
votes
0answers
11 views

How do I get the proper p-value of a time series average of regression coefficients?

I have run cross-sectional regression on the returns of 50 companies on 16 regressors, for 128 days. The regression output looks something like this: ...
0
votes
1answer
33 views

A question about co-variance of two linear combinations

The variance of two linear combinations can be expressed as $\rm{var}({{\bf{a}}^{\rm{T}}}{\bf{X}})={{\bf{a}}^{\rm{T}}}\rm{var}(\bf{X}) {\bf{a}}$ where ${\bf{X}}$ is a vector of random variables and ...
1
vote
0answers
18 views

Some statistical (learning) issues

im reading about statistical learning (Trevor Hastie, Robert Tibshirani, Jerome Friedman The elements of statistical learning) and for some reason it seems to be trivial that $E[XX^T]$ is non-singular ...
0
votes
1answer
33 views

How do I test whether non-zero numbers in a long datastream are clumped more than chance?

I have a long string of integers between 0 and 9. Most of them are 0, but some are not. I want to know if the non-zero ones "clump" together - if their distribution is such that it is statistically ...
0
votes
0answers
15 views

Moments of Multivariate Normal Distributions

I have two questions. Suppose we have two multivariate normal distributions $X \sim N(\mu,\Sigma)$ and $Y\sim N(c\mu,\Sigma)$ where $0<c<1$ is a constant, $\mu$ is a vector and $\Sigma$ is a ...
0
votes
1answer
36 views

Can I represent or write continous density function in the following way?

If $$f(x)=\cases{1 &for $0<x<1$\\0& for $x=0$\\ 0 &for $x=1$}$$Can I say f is a density function of a random variable on [0,1]. I mean to say is it correct way to write or define the ...
1
vote
1answer
22 views

Use of binomial coefficient in order statistics

Appreciate all your help with the following: Let $U_{1}, ... U_{n}$ be n independent and uniformly distributed on (0,1) random variables. Let $U_{(1)} \geq U_{(2)} \geq ... \geq U_{(n)}$ be an ...
0
votes
0answers
30 views

What is the maximum entropy distribution over all integers (ie. including negative ones) with fixed mean and variance?

I know that the maximum entropy distribution with over the non-negative integers fixed mean is a geometric distributions. However, I cannot find conclusive information about what are the maximum ...
1
vote
2answers
28 views

Question regarding probability and dice rolling [closed]

Four red dice and six blue dice are rolled. Assuming that all ten dice are fair six sided dice, and rolls are independent, what is the probability that exactly three of the red dice are even, and ...
0
votes
1answer
15 views

How principal component analysis ensures component orthogonality when using zero co-variance as the restriction to maximize variance?

I am currently learning the mathematics behind PCA and I found when PCA maximizes variance to find out the 2nd, 3rd, ... components, it uses zero co-variance as the restriction, as shown below, ...
2
votes
0answers
27 views

Average Cost of Obtaining in game Item

I know this will sound like a trivial maths problem, but recently I've been playing a game in which you can pay 5 in game gems to get a Rare (R), Super Rare (SR), and Ultra Rare (UR) characters ...
1
vote
1answer
18 views

How to refine population statistic when more data is available

Suppose I have two pieces of data about two populations. The first piece of data is the national accident rate, denoted A The second piece of data is the national safety rate, a related, but not ...
1
vote
1answer
23 views

What is the notation (if any) for series probability inclusion?

In statistics, what is the notation to use for an event $A$ in $B$ in $C$ in $D$, etc., where the series may continue for a large number of events? The following works for a few events: $$A\cap B\cap ...
0
votes
0answers
14 views

calculating the risk function $max(\bar{X},2)$ under Squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I calculate the risk function $max(\bar{X},2)$ under Squared error loss function
1
vote
2answers
23 views

Probability of birthdays on a specific day problem

Here's a basic stats problem that is bugging me. If there is equal probability of someone being born on any day of the year, then what is the chance that a randomly chosen person would have been born ...
0
votes
1answer
26 views

Finding the standard deviation without anything except the mean

My statistics professor gave us this question on our last exam and to solve for the standard deviation he just took the square root of the mean, is this correct? Q: An average of 15 aircraft ...
0
votes
0answers
20 views

Maximal Bayes error for two classes

What is the $maximal$ Bayes Error for a classification problem of $two$ classes (each a distribution with known parameters) and class prior $\alpha$? Note: for two classes and $\alpha=\frac{1}{2}$ ...
0
votes
0answers
22 views

Dirichlet parameters influence on samples

If I draw samples from a Dirichlet distribution with parameters $[0.1, 0.1, \dotsc, 0.1]$, I get samples like multinomial, but if I set parameters to $[100, 100, \dotsc, 100]$, I'll get uniform ...
-1
votes
0answers
30 views

Inequality involving Probability of supreme and supreme of probability

Consider random vector $y$ with support $\Lambda\subset\mathbb R^m$ and a deterministic mapping $f(\cdot,\cdot):\mathbb R^n\times\mathbb R^m\longrightarrow\mathbb R$. For an arbitrary but compact set ...
1
vote
2answers
18 views

Calculating a normal distribution with a sample size?

the sample of $n=25$ is what is throwing me off. I have no clue what to do with it. Given a normal distribution with $\mu=101, \sigma=25$, and given you select of $n=25$ $A.)$ $P(\overline{X} ...
0
votes
1answer
34 views

Urn has theta black balls, N - theta white balls. Sample of n balls, let Y denote # black balls in n. Show (N|n)Y is MOME of theta.

First I am confused on the notation. (N|n)Y, is that saying N given n? Then, I know we need to find the kth raw moment and set it equal to the kth sample moment. But I am not even sure where to ...