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
1answer
15 views

Probability question with a radio competition

I'm quite new to statistics and I'm going through a few exam questions but I am a bit stuck on this one: ...
0
votes
2answers
29 views

Probability of incorrectly spelling a word

I'm currently trying to teach myself Statistics and have an exam question that I need a bit of help on: ...
2
votes
0answers
6 views

Derive the Hat Matrix to map actual response to estimated resposne

In order to measure the quality of a regression we can calculate the Hat Matrix. Using it we can estimate the response variable as if we used the predictor variables to regress them. For linear ...
0
votes
1answer
8 views

loss function similar to normal density

let $$L_\epsilon(x,p) = -\frac{1}{\sqrt{\epsilon}}\exp\left\{-\frac{(x-p)^2}{\epsilon}\right\}$$ be a loss function. given a random variable $X$ with density $f$ (possibly restricted), the risk ...
1
vote
0answers
12 views

If I know the mean, median, mode, and so on how do I determine the standard PDF that most fits

I have been playing a game and it drops currency with a drop rate that has an expected value of $1/750$ and nothing else. I was able to keep track of 3000 drops and times between them. I can then find ...
4
votes
2answers
127 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and $\...
2
votes
3answers
19 views

Tricky permutations question

There are 8 buckets, each bucket is a different color (for simplicity, let's label the colors A, B, C, D, E, F, G and H; if you like: Aqua, Brown, Cyan, Diamond, Eggshell, Fuchsia, Green, Hot-pink). ...
0
votes
1answer
69 views

Measuring the degree of convergence of a stochastic process

Consider a set of random variables $(X_1,X_2,X_3,...X_k)$ that are i.i.d. $Bernoulli(p)$ While I do not know $p$, I can estimate it using $$ Y(k)=\frac{1}{k}\sum_{i=1}^k X_i $$ Notice that $Y(k)$ is ...
0
votes
0answers
20 views

MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased? Does there exist an unbiased estimator of $P\{X_{i}=...
0
votes
0answers
8 views

Existance of a UMVUE

$ \{X_{i}: 1\leq i \leq n \} $ is a random sample, i.i.d $ N(\mu, 1) $ with $ \mu $ unknown. For a fixed $ x_{0} $, does there exist a UMVUE for $ \phi(x_{0}-\mu) $, where $ \phi $ denotes standard ...
0
votes
0answers
13 views

The relation of correlation coefficient of the sum of two vectors.

Does the correlation coefficient of the sum of two vectors between the correlation coefficient of each of them. Suppose I have three vectors $x_1,x_2,x_3$. The correlation coefficient of $x_1$ and $...
0
votes
3answers
38 views

Normal distribution

can anyone help me calculate $E(Z^4)$, $E(Z^3)$ for $Z\sim N(0,1)$? I know that $Z^2\sim \chi^2(1)$ then $E(Z^2)=1$, $Var(Z^2)=2$. Thank you.
0
votes
1answer
174 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
0
votes
1answer
29 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
1
vote
2answers
26 views

Why is it true that $S'(t)/S(t) = d log(S(t)) / dt$?

I came across this identity in derivation of the hazard rate in survival analysis.
0
votes
2answers
35 views

given a graph of density function, what can we conclude about expected value

given the following graph (the density function), what can we conclude about the expected value? I got stuck a little bit with that question and I would appreciate your help! I found out that C must ...
0
votes
2answers
51 views

Why is the probability density function of a normal distribution exponential? [on hold]

I came across this while self-studying for a probability course but I still don't quite get the rational behind it. Would appreciate it if someone could provide some intuitive explanation or rigorous ...
0
votes
0answers
12 views

Calculating RMS score difference

I have a problem to figure it out how they calculated the RMS score difference between winning and losing team in the article (https://arxiv.org/abs/1503.03509). win.avg = 102.1 points, loss.avg = 91....
0
votes
3answers
35 views

given the following CDF, find the expected value

I got stuck at the middle of the question. would appreciate your help. first of all, given the CDF as follows, I had to find parameters $a$ and $b$ such that the CDF is a function of a continuous ...
-1
votes
0answers
16 views

Predicting transfer value of a Football player given performance scores [on hold]

I just recently landed my dream internship at a football statistics company and I am eager to impress. I have an excel spreadsheet of every player in the major leagues along with the minutes they ...
1
vote
0answers
115 views

Multivariate convergence in distribution

Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$. Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$. Prove that convergence in Distribution of $$ \sqrt n \left(\...
2
votes
1answer
39 views

Relation between these two series

Assume a constant $\alpha$ and $N$ positive integers $\{n_1,...,n_N\}$. What is the relation between $\frac{N\alpha}{\sum_i{n_{i}}}$ and $\sum_i{\frac{\alpha}{n_{i}}}$ when $N\rightarrow\infty$? $N$ ...
0
votes
1answer
672 views

variance validation

The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean $μ = 74$ and a variance $σ^2 = 8$. Would you still consider $σ^2 = 8$...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
2
votes
1answer
21 views

Unusual graph measure

Integrated information theory of consciousness is a complex mathematical model of information transfer in neural networks. Some of its conclusions are obvious: neither fully disconnected nor the ...
0
votes
0answers
10 views

Calculating rate of time consumption based on NBA player impact estimate

I'm building a game. I'm looking for help calculating the rate of consumption based on a player's impact estimate (PIE). PIE measures a player's overall statistical contribution against the total ...
2
votes
1answer
17 views

Expectation of absolute sum of squared normal distributions

Let $u_i$ be a standard normal distribution for all $i$. All $u_i$'s are independent of each other. I want to compute the expectation of: $$| \sum_i u_i^2 \lambda_i |$$ Where $\lambda_i$ is real ...
1
vote
0answers
28 views

l1 regularization mathematical explanation [on hold]

I somewhat understand what l1 regularization is, however, the mathematical formula and how to use it are confusing me. I'm not really sure what a regularization term is and how I could apply it to a ...
0
votes
0answers
20 views

MLE of heteroscedastic model?

Given the regression model where our and are identically and independently distributed. I'm trying to find the MLE B-hat and the unbiased estimator sigma-hat^2. I haven't dealt with any models in ...
1
vote
0answers
48 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
2
votes
1answer
702 views

Real life math to explore/solve [on hold]

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
0
votes
3answers
2k views

Easy GRE question: Statistics

I'm not sure how to set this statistics problem when they give me a group of arbitrary values. Can someone help? A group of 20 values has a mean of 85 and a median of 80. A different group of 30 ...
1
vote
2answers
62 views

Variance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
4
votes
1answer
56 views

Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
0
votes
3answers
39 views

Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
1
vote
2answers
34 views

Why does a null-hypothesis have to have a definite value?

In hypothesis testing, why does the null hypothesis (H_0) have to have one defined value?
1
vote
1answer
33 views

Interquartile range to find out outlier & get perfect Standard deviation

I have one population dataset - 200, 330, 675, 999, 1200, 3000, 25000 For this dataset IQR = 3000 - 330 = 2670 Also we will ...
0
votes
0answers
19 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
1
vote
0answers
22 views

Statistical calculation of value of coins in a box

I woke up from a dream today that made me consider the following scenario: A grocery store has an electronic donation box. Good Samaritans slide coins into the donation box, and the donation box ...
0
votes
0answers
13 views

Trouble with Bayesian Hypothesis Test Equation

A passage from Wasserman's All of Statistics: The Bayesian approach to testing involves putting a prior on $H_0$ and on the paramater $\theta$ and then computing $\mathbb{P}(H_0 \mid X^n)$. ...
0
votes
1answer
33 views

Showing sum of squared residuals is zero?

I have the model $$y_i = B_0+\sum\limits_{i=0}^pB_kX_{ik} + e_i$$ I'm looking to show the sum of squared residuals is zero if $p = (n-1)$. I have tried expanding it quite in depth and I haven't been ...
0
votes
1answer
54 views

Deriving MLE for covariance matrix using Robbins-Monro

I'm having some trouble completing exercise 2.37 in Bishop's Pattern Recognition and Machine Learning text. I'm not reading this text as part of a course, so this is not a homework question. Here's a ...
0
votes
2answers
24 views

Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
-3
votes
0answers
36 views

definition of a chi-square-distributed random variable in terms of independent N(0, 1)-variables [on hold]

What is the definition of a $\chi^2$-distributed random variable in terms of independent $\mathcal N(0, 1)$-variables? What is the relationship of a $t$-distribution with a standard normal and $\chi^2$...
0
votes
2answers
486 views

Random sampling of 100 data points from a binomial population

This is my question I need to answer: Generate a random sample of 100 data points from a binomial population B(40, 0.4). I'm pretty sure that n=100 and my mean is 40 and standard deviation is .4, ...
1
vote
1answer
39 views

Showing Residual Sum of Squares for Multiple Linear Regression is 0

Problem: I have the linear regression model: $y_i=\beta_0+\sum_{k=1}^p \beta_kx_{ik}+\epsilon_i$ where $\epsilon_i\sim N(0,\sigma^2)$, for $i = 1,2,\ldots ,n$. I want to prove that the residual sum ...
1
vote
1answer
579 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
0
votes
1answer
5k views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
0
votes
0answers
17 views

How to best compare two different time series with different frequencies

Lets say I have two time series $X_t$ and $Y_{t,q}$. As an examples, lets say $X_t$ is a series that measures year over year changes in the level of output of a good (say number of widgets). So $X_t = ...
0
votes
1answer
560 views

Show that -log transformation of Pareto distribution is exponentially distributed.

Question: Given that y is distributed as: $$ f(y; \theta) = \theta y^{(\theta-1)} $$ $$0<y<1 , \theta>0$$ If Z = -log(Y), show that Z has an exponential distribution.(ie $E(Z) = 1/\theta$)...