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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74 views

Two dice thrown, one comes up 6

If my friend throws two dice, and covers them up, but I see that one of them was a 6, what's the probability that they were both 6s given this knowledge? I'm under the impression that the answer is ...
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2answers
23 views

Probability of a draw given 15 pairs of objects

Given 15 pairs of objects, what is the probability that you will draw exactly one of your desired object if you were to take 3 out of the 30 total objects? If possible, work this out by not using the ...
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1answer
9 views

Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
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1answer
13 views

How to extract an individual's (normalized) contribution from a group?

To make this concrete assume you have two teams playing against each other. Each team can only have n players on the field, and may be substituted for a different player at different times during the ...
0
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1answer
24 views

PLotting a Gaussian in python

so I am trying to plot a histogram of my data and I seem to be a little confused here. I am using matplotlib in python. Here is the code from their website: ...
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1answer
28 views
0
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1answer
25 views

Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
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0answers
24 views

Statistics: How would I correlate many variables to a few coefficients?

I'm trying to predict the "strength" vs "tempertature" and "time" curves of some chemical compounds as a function of the concentrations of their component substances. I have 20 different substances. ...
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1answer
59 views

Non-linear least squares with two dependent variables

I have data in the form $(t_i,x_i,y_i)$, i.e. position in 2D as a function of time. I have non-linear equations which I want to fit to the data. They give me a position $(X,Y)$ as a function of time ...
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0answers
14 views

Is there a bound on largest eigenvalue for covariance matrix of discrete random variable?

I have a random variable $Z=(Z_1,\ldots,Z_p)$. Each component can take values in {-1,0,1}. Is there a way to bound the largest eigenvalue of Cov(Z)? Actually, I have a latent multinormal variable ...
2
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1answer
34 views

How do you test if the observed data have the same mean?

This seems like it should be straightforward but I am at a loss. How does one test the following hypothesis: the observed data have the same mean (or, alternatively, the observed data have the same ...
0
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0answers
24 views

Autocovariance Function of this $AR(1)$ Process

Consider the $AR(1)$ process given by $(1-0.6B)(X_{t} - 3) = a_{t}$ where $a_{t} \sim WN(0,1)$ and $B$ is the backshift operator ($X_{t}B = X_{t-1}$). We can rewrite the process in the more ...
0
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1answer
35 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
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0answers
11 views

Modeling Counts with Small Number of Observations

I have a large data set that contains $5$ different fields. The fields are ...
2
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2answers
40 views

Neyman-Pearson lemma. Doubt on the text of the lemma

In my book: $\mathbf{X}=(X_1,\ldots,X_n)$ $f(\mathbf{x})$ is the joint density, where $f$ is either $f_0 \text{ or } f_1$. Suppose we want to test $H_0: f=f_0$ or $H_1: f=f_1$. The test, whose test ...
0
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1answer
8 views

Modelling a normal-like single-ended random variable

I am trying to model a of (normal-distribution-like) discrete random variable using the normal distribution. This is what I understand so far: First, I approximate the mean of the normal ...
0
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1answer
19 views

Testing a hypothesys about a survey?

A survey of $61, 647$ people including questions about office relationships. Of the respondents, $26$% reported that bosses scream at employees. Use a $.05$ significance level to test the claim that ...
1
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1answer
28 views

Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
1
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2answers
31 views

What is the expected value of this sum (drawing slips from a bag)?

Suppose there are thirteen slips in a bag, labelled 1-13. If I draw a 10, 11, 12, or 13 then I stop adding to the sum and return the sum. If not, I add the number I draw to the current sum and place ...
2
votes
1answer
93 views

Simplify this summation

$$\sum_{k=5}^\infty{{k-1}\choose{k-5}}\frac{k^3}{2^k}$$ I can't seem to simplify this sum. I get to a certain point then I get stuck, I know there must be some sort of trick to simplify it but I am ...
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2answers
9 views

calculating stdev from a previous stdev [on hold]

I am writing a program that tracks performance, is there a way to calculate stdev when the only data i have is the new value, old mean, old N, old stdev? eg. ...
0
votes
1answer
24 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
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1answer
54 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
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0answers
15 views

Choice of the ridge parameter using leave-p-out estimation of the risk

The context of my question is the kernel ridge regression (in the field of Statistics). The question is to find a closed formula for $\lambda^{*}$ \begin{align*} \lambda^{*}=\underset{\lambda >O ...
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0answers
9 views

Sparse data and covariance matrix computation

Background I am trying to apply Gaussian Discriminant Analysis (GDA) on the MNIST dataset of hand-written digits, with 10 classes for 10 digits. In this dataset, each point is a vector of 784 ...
0
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1answer
23 views

Can't see how this equality that involves the gamma function holds

$$\frac{1}{\sqrt{2\pi}}\sigma^m \gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) ...
1
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1answer
18 views

MSEP error derivation

I have random variable $X$ and set of observations $\mathcal{D}$. Let $\widehat{X}$ be an estimator for $E[X|\mathcal{D}]$ and predictor for $X$. Mean square error of prediction MSEP is defined as $$ ...
3
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1answer
32 views

Math formulas on Clustering

I am currently studying Clustering in Machine Learning. I have found a document regarding guessing the right number of clusters. I am reading the first part of it, having difficulties in understanding ...
0
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4answers
61 views

Min and Max of $ f(x,y)=\frac{x-y}{a-x-y}$

I'd like to find max and min of $$ f(x,y)=\frac{x-y}{a-x-y}$$ where $0\le x<y\le a/2$. Any one can suggest? Thank you
2
votes
1answer
37 views

Simple way to derive formula from dataset?

I'm not too good at math but I'm going to give this ago anyway and try and explain. Say I have a dataset, where each object in the set (not too sure about my correct use of terminology!) has $4$ ...
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1answer
31 views

Projection equation

I'm a programmer, not a math expert or statistician by any means, but my organization wants a page in our admin console that displays a projection of how many registrations we can expect to see based ...
2
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1answer
32 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
2
votes
1answer
24 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
1
vote
1answer
16 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
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0answers
20 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
2
votes
1answer
25 views

What do I need in order to draw conclusions from this data?

I have three techniques, called A, B and C. Each can be used independently when trying to perform four related tasks (Tasks 1, 2, 3 and 4). I have run lots of tests, and tried all combinations of each ...
3
votes
1answer
35 views

Is it possible to determine the average and median if we only have the upper part of a bell curve?

Practical example: http://dota2.com/leaderboards shows the ~800 best solo ranked match-making scores in an Elo rating system. The lowest possible score is 1. There were 9318362 players last month, but ...
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0answers
10 views

variance-covariance matrix of 2sls in homoskedasticity and heteroskedasticity

How the variance-covariance matrix of two stage least square estimate in homoskedasticity or heteroskedasticity? The answer in homoskedasticity is $\sigma^2(E[xz'](E[zz'])^{-1}E[zx'])^{-1}$ and in ...
0
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0answers
21 views

Finding the underlying model from data

Let's say I have "k" data points in "n" dimensions. I also have "k" results of an unknown function of this data. Is it possible, simply given the data and the results from the function, to guess the ...
0
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0answers
13 views

L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
1
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1answer
21 views

Expected value of $(\overline{X} - 5)^2$?

$X_1, \ldots, X_9$ are 9 random samples from a N(5, 9). I am looking for the distribution ofExpected value of $(\overline{X} - 5)^2$. $$E[(\overline{X} - 5)^2] = E[\overline{X}^2 - 10\overline{X} + ...
0
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1answer
37 views

Do doctors have a shorter lifespan than the general population - Null Hypothesis

Do doctors have a shorter lifespan than general population with this data: Known lifespan for US subject in last 2 decades = 70.1, with a standard deviation of 7.6. A sample from 51 doctors from the ...
2
votes
2answers
62 views

Why is this expectation true?

Working with Rao-Blackwell, this came up: $$E[2X_1 \mid \max(X_i) = t] = 2\left(\frac{1}{n}t + \frac{n-1}{n}\frac t 2\right)$$ Where X are uniform(0, $\theta$). What are the intermediate steps? I'm ...
0
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1answer
24 views

Is there a significant difference in population percentages

The question is, "If an 85% confidence interval (CI) for the difference between the population of A Students (Pa) and B Students (Pb) is -15.8% < Pa-Pb < -7.9%, is there a significant ...
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0answers
14 views

Underperformance of an NCAA team

If we played a game where a NCAA team was worth 100 for winning, 50 for second place, 25 for QF, 10 for R16, 5 for R32 and 0 for R64, and we had our expected values are based on the oddskmakers ...
6
votes
3answers
135 views

Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
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0answers
13 views

Bias of the MLEs for the two-parameter Weibull distribution

Is it possible to obtain a formula for or an equation on the exact bias of the MLE-vector for the two-parameter Weibull distribution (both parameters unknown). I've read papers offering Monte Carlo ...
0
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1answer
46 views

Assigning $\pm 1$ values to the edges of a complete graph

I read this sentence in one combinatorics book. In graph $K_{100}$, there is a possible way to assigns number (value) from $\{+1,-1\}$ to each edge, so that the sum of all edge values connected to ...
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0answers
26 views

How to find mean and median from histogram

Solution for finding mean : The problem faced when i saw a video to evaluate the mean https://www.youtube.com/watch?v=vMrc6dP8pCo According to the video, the lecturer said that, we can take ...
4
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0answers
61 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...