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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Is there an opposite of Chi Square test?

I know basically nothing about statistics. I find another post about chi-squared test, but it only tells me whether given values not fit expected distribution, but not when they do fit: Is there a ...
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65 views

Uniform Sampling over Convex Polytope (not full-dimensional)

I want to simulate a uniform distribution on a convex polytope that is not full-dimensional for optimization purposes (to generate random points on the set I want to minimize over). The polytope is ...
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13 views

Converting normalised values into original

I have a normalisation formula as follows, which takes a list of numbers, such as $1,2,3,4,5,6,7,8,9,10$, and returns the normalized values which guarantees that $\tilde{x_i} \in [0,1]$. ...
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What elementary statistics (eg correlation) can help me with an Artificial Intelligence program I'm writing?

I am writing a crude Artificial Intelligence langauge/conversation computer program in java for a first year community college creative final assignment. I know some basic Statistics / Finite ...
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hi, for an independent event, like flipping a fair coin does Pr(A|B) always equal to Pr(B|A)?

for an independent event, like flipping a fair coin does Pr(A|B) = Pr(B|A)? Example You flip a fair coin, independently, three times, Event A. The first flip results in heads Event B. The coin ...
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79 views

Sampling Distribution; Statistics. Please verify my answer

Professor earns average $ \text{\$} 65,500$ per year with standard deviation of $\text{\$}3,500$. Random sample of $64$. a. Describe sampling distribution of sample mean $\bar{x}$ of average ...
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25 views

not consistent estimator

Let $(X_1;X_2;...;X_n)$ be a sample from exponential distribution with the unknown parameter $\lambda$ > 0. Show that the statistic $T_n(X_1;X_2;...;X_n) = nX_{1:n}$ is not consistent estimator of ...
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8 views

Fill chart using known final answers [duplicate]

so all I know is that: n=100. i have 5 departments within the numbers: 0-1000. md = 500. avg =490. lower decile=200. upper quarter=600. I really don't know how to use the formulas of each if I don't ...
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32 views

Bradley-Terry model in sport

I was reading about Bradley-Terry model for paired preferences in Categorical Data Analysis, by Alan Agresti because I am developping a statistics module in a project. If I take this table from the ...
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1answer
44 views

Should you choose highly owned or little owned players in fantasy sport

Here's the situation: It's a fantasy soccer game where players score points for my team based on their actual performances on the pitch. I have a team of 11 players and their is no limit to the ...
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30 views

What is the correction factor in a generalised log-linear model?

I am following the time-independent bivariate poisson model on a wikipedia page on predicting football matches, and do not understand what is the correction factor. It is mentioned in this section ...
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43 views

Fill chart using final data

so all I know is that: n=100. i have 5 departments within the numbers: 0-1000. md = 500. avg =490. lower decile=200. upper quarter=600. I really don't know how to use the formulas of each if I don't ...
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18 views

Sufficient statistics for $\mu_1 -\mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is ...
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1answer
42 views

how to calculate standard deviation

I have the following chart. How do I calculate the standard deviation? I've tried to do that: $x=$ $[2^2*30+6^2*35+(9.5)^2*10+13.5^2*20+18^2*5)] \over 100 $ $-(7.25)^2$= 22.912 -> wrong answer. what ...
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40 views

Visual representation of type II error

I doing some stats exercises, in which i have to calculate the Type II error for an normal distribution. I understand the concept of it, that you accept $H_0$ even though it's not the true means, ...
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87 views

Jensen’s inequality

Prove the following by using Jensen's inequality: $$\sum_{i=1}^m \exp(\theta^\top f_i) \geq \exp\left(\theta^\top \sum_{i=1}^m \alpha_i f_i - \sum_{i=1}^m \alpha_i \log \alpha_i\right)$$ where, ...
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1answer
17 views

Goodness of Fit expected value

If I have a classical experiment with peas, where in the experiment there are hybrids produced in such a way that probabilities of observing the different phenotypes are: Round and Yellow: 9/16 ...
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1answer
21 views

what is one basic/intermediate regression analysis standard textbook that is math intense

What is one basic/intermediate regression analysis standard textbook that is math intense with proofs/derivations? Also, i need that one to be comprehensive yet the diffculty is suitable for self ...
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1answer
41 views

Projection of $\log(x)$ on $(x,x^2)$

For a positive random variable $X$, is it true that $\mathrm{cov}(X,\log(X))>0$ and $\mathrm{cov}(X^2,\log(X))<0$, where $\mathrm{cov}(X,Y)$ denotes the covariance between any $X$ and $Y$? The ...
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32 views

Adding a factor to a ranking?

I have a ranking of 10 items from best to worst. Let's assume that the best is ranked 1 and the worst is ranked 10. Each item is ranked according to some rules that we cannot know so all we get is the ...
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76 views

Sum of n iid random variables

Let $X_1,X_2,\ldots,X_n$ be iid poisson random variables with mean $\lambda$ , then it can be verified using mgf that the sum $S=\sum\limits_{i=1}^n X_i$ is also poisson with mean $n\lambda$. ...
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42 views

what should the null hypothesis be

A group of 50 complete a national fitness test and get a mean score of 80 out of 100. The national average is 72 with standard deviation 6. Can we conclude the group of 50 is fitter than the national ...
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1answer
44 views

mean of random variables

Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ a discrete random variable and $g : \mathbb{R} \rightarrow \mathbb{R}$ a random variable. I can't ...
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48 views

Matrix form for Weighted Least Squares

If we have the following weighted least-squares regression, with $\hat{\beta} = (X'WX)^{-1}X'WY$ How can we express the squared errors, MSE and the fitted values in matrix form? These are the OLS ...
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82 views

Expected Value of product of Ito's Integral

Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit? For example: $$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$ I only ...
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1answer
58 views

Method of moments: unbiased estimator for small samples

When we want to find an estimator of a parameter of a distribution function using the method of moments (order 1), is it true that the estimator found is unbiased ? I am pretty sure that this is not ...
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147 views

Partial proof for correlation coefficient formula?

I've been working to prove the formula for the correlation coefficient, since asking my last question yesterday (Meaning of denominator in correlation?). If this post in any way violates any ...
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3answers
45 views

Expected value of absolute value of trades

Suppose each day for $365$ days, we flip a coin. If it lands heads, I get \$10. If it lands tails, I lose \$10. What is the expected value of the absolute value of the amount of money I have (or owe) ...
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52 views

The estimator of the first beta distribution parameter

Let $f(x;\theta)$ be the probability density function of the beta distribution with parameters $\theta$ and 2, i.e. $$f(x;\theta) = \theta (\theta+1) x^{\theta-1}(1-x), \ x\in(0,1), \ \theta \in ...
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solving and predicting complexity of a statistical model

How difficult/time consuming would it be for a professional mathematician to model a temporal probability distribution of when an event will occur when the temporal history of that event occurring is ...
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34 views

Generating a data set that approximates another given random variable

One of the question my friend was asked in technical interview : The probability density function for a normal random variable with μ = 0.6 and σ = 0.7:​ Below is the code to generate the ...
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140 views

Is this a simple change of variables, or is there something else going on?

This is a question from a past statistics class, and I'm a bit puzzled by the first few steps in its solution. Suppose $(X,Y)$ is uniformly distributed on the triangle with vertices $(0,0), (1,1), ...
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1answer
53 views

Which one is bigger $D(P\Vert Q)$ or $D(Q \Vert P)$?

In general the Kullback-Leibler divergence is asymmetric. If $P$ and $Q$ are two distributions $D(P\Vert Q) \ne D(Q\Vert P)$. However, I was wondering if there are situations where we can say which ...
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107 views

Distribution of $\sqrt{X_{1}^{2} + X_{2}^{2}}$

Suppose $\mathbf{X} \sim \mathrm{unif}(B^{2})$. Then $\mathbf{X}^{2} \sim \mathrm{Dirichlet}_2 \left( \frac{1}{2}, \frac{1}{2}; 1 \right)$. I want to find the distribution of $R = \sqrt{X_{1}^{2} + ...
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234 views

Meaning of denominator in correlation?

I can't quite grasp the meaning of the denominator in the correlation coefficient. $$\frac{\sum(X - \bar X)(Y-\bar Y)}{\sqrt {\sum (X-\bar X)^2\sum(Y-\bar Y)^2}}$$ What exactly am I dividing with, ...
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1answer
51 views

Question about correlation in statistics

A Feb 2008, how would the correlation change reported a study on the distance students lived from campus and average GPA. Here is a summary of the result. Residence / Avg. GPA Residence hall / ...
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Question about correlations in statistics

x: 1,2,3,4,9,10 y: 12,2,3,5,9,11 What feature of the data is responsible for reducing the correlation to this value despite a strong straight-line association between x and y in most of the ...
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26 views

How to account for both between-subjects and within-subjects covariables?

I have a data set I'm trying to analyze and I can't figure out how to include the two different kinds of covariables in my analysis. Without the covariables, the analysis isn't too complicated. It's ...
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1answer
170 views

If $U$ is uniformly distributed on $S^{2}$, then its first component is uniformly distributed on $(-1,1)$.

Assume $\mathbf{U} = (U_{1}, U_{2}, U_{3})' \sim unif(S^{2})$. How would I show that $U_{1} \sim unif(-1,1)$? I don't know if I'm confusing myself, because I can't see this as being true for higher ...
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51 views

Exponential Confidence Interval

It is known that, for large $n$: $$\sqrt{n}(\lambda\bar{x}-1) \sim \mathrm{Normal}(0,1)$$ With this approximation, show that the 95% confidence interval for $\lambda$ is: $$\frac{\sqrt{n} - ...
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47 views

convergence in probability of mean of sequence $X_{k+1}=\beta X_k+\epsilon_k$

Let $X_1=0$ and $X_{k+1}=\beta X_k+\epsilon_k$ with $\epsilon_k$ iid normally distributed, $\epsilon_k\sim N(\mu,\sigma^2)$. Let $\mu$ and $\sigma^2$ be fixed. How can you show the mean of $X_k$ for ...
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730 views

Central Limit Theorem for exponential distribution

Suppose that $X_1$ ..... $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, ...
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15 views

CDF of time spent to complete two Jobs with time that follows a Poisson distribution

Suppose that we have a single core CPU that start to compute two jobs at time t=0. Let T be the time spent by this CPU in order to complete these two jobs, what will be the CDF of T in this two ...
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Calculated Weighted Standard Deviation from Standard Deviation and Weight Only

I have a table of unweighted means and standard deviations for some subsetted data, along with how many points of data contribute to each mean (the weight). All the data points are from the same data ...
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1answer
52 views

Expected number of passengers for the flight

One of the interview questions that I was asked was : The small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.1, independent of ...
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91 views

Analyzing/Modeling DynamoDB Usage

I'm seeking the correct mathematical tools to analyze/model a real world process for the purpose of optimizing resource allocation. I want to analyze consumed throughput of a AWS DynamoDB table and ...
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42 views

Bivariate sampling for distribution expressed in Sklar's copula theorem?

In the univariate case, one can easily sample a distribution via random numbers $u\sim[0,1]$ and plugging into $F^{-1}(u)$. I have a bivariate distribution constructed via Sklar's theorem on Copulas: ...
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34 views

Cumulative distribution function .

I have a random variable $X_n = 1+ \frac{1}{n}$ , I don't understand completely why the cumulative probability function $F(x)$ is given by $$F(x) = 0 , t < 1+\frac{1}{n}$$ and $$F(x) = 1 , t \ge ...
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23 views

Help relating gaussian to chi-squared distribution

I am having trouble finding a simple layout/documentation for the chi squared distribution. From what I understand the chi squared distribution is just: Where "v" is some strength parameter. Now, ...
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39 views

Probability Urn Group Problem

my group and I are having trouble figuring out how to do this. For some reason I have an urn that contains 10 coins. 3 of the coins are blue on one side and red on the other, 3 of the coins are blue ...