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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Explaining probability theory versus statistics

I'm not sure whether this question was asked before, but it's hard to search because of lots and lots non-descriptive titles like "statistics and probability". The context: There is an anecdote I ...
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12 views

Statistics Confidence Interval Estimators

I'm working on this problem and don't know how to approach it: Could anyone help steer me in the right direction? Thanks
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58 views

Utility of Probability Generating Function .

The utility of Probability Generating Function , how far known to me , is basically to generate PMF uniquely (what all the popular books of probability have written ) . Now , PGF is constructed with ...
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13 views

Computing a threshold value for paired data

I have a dataset of vector pairs; for each pair ($x_1$, $x_2$) I have two values $V$($x_1$,$x_2$) and $U$($x_1$,$x_2$) (two metrics). I also know a threshold value $U^*$ such that any pair ($x,y$) ...
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2answers
95 views

What does (1 + 5%) mean

(I'm a software developer so excuse me) I'm building an application for a client and one of the formulas that has been provided in the spec is ...
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18 views

Statistics Parameter Estimators

I've been struggling to figure out how to do the following question: Can anyone help me figure this out? Thanks.
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47 views

Determine if a function is a gamma distribution

Focusing on the last 2 equations... Why is $$\frac{\lambda^{n+a-1} \exp\{ -\lambda(\sum x_i + b) \}}{\color{red}{(\frac{1}{\sum x_i + b})^{n+a}} \Gamma(n+a)}$$ I am thinking the part marked red ...
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31 views

Maximum Likelihood Estimate and Bayesian Estimation. $f(x_1,…,x_n)$ vs $f(x_1,…,x_n | \theta)$

In part (a) of the question (in image below), $$f(x_1,...,x_n) = \lambda^n e^{-\lambda \sum x_i}$$. Then in part (b) I am supposed to use the bayesian method. The way I understand it, I need to find ...
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60 views

Given a score, work out the portion of the bell curve below that score.

I am writing an application which will monitor stock prices and alert the user of abnormally large price changes. Given a set of previous price changes (fitting into a bell curve) and a new ...
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1answer
83 views

Book for probability and various probability distribution functions.

Please suggest a book/books where i can understand Probability theory (with lots of example and solution) examples on permutations and combinations. list of all probability distribution functions, ...
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23 views

Suggesting a UMVU for a poisson distribution

The number of clients arriving at a certain bank in an hour poissonically distributed with $\lambda$. Suggest a UMVUE according to the following 4 observations: $12,25,18$ and $27$. I know that a an ...
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25 views

Normalized cross-correlation in detail

I'm trying to implement a normalized cross-correlation algorithm but I don't get what in fact is this measure. What confuses is the wikipedia definition: $\frac{1}{n} \sum \frac{(f(x,y)- ...
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58 views

Variance of an unbiased estimator of central moments

Some days ago I asked this question: Unbiased estimators for the moments of 2 not independent random variables Now, using the some framework of the previous question, I'm facing with the problem of ...
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23 views

How to extract covariance (error) matrix of independent from covariance matrix of dependent parameters?

Suppose I have a covariance (error) matrix C_xyz for system with 3 parameters: x, y, z. Then it appears that ...
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1answer
43 views

Why does $p(X\;|\;Y) = \displaystyle\frac{p(Z,X\;|\; Y)}{P(Z\;|\;X,Y)}$?

I'm reading about expectation maximization and on one point in my paper it is said that: According to probability theory: $$p(X\;|\;Y) = \displaystyle\frac{p(Z,X\;|\; Y)}{P(Z\;|\;X,Y)}$$ Where $p$ ...
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41 views

Mean of a binomial distribution

During an election, a proportion $p$ of voters choose candidate $A$, while a proportion $1-p$ choose candidate $B$. A number of $n$ voters have been polled before the election. Let $X_i$ be $1$ or $0$ ...
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2answers
147 views

expected value hack-a-shaq problem

I am having trouble feeling confident i got the right answer here. i am not sure about the free throw calculation. my initial feelings were to do 80*.466 + 40*.466 + 0*.466 for the expected value of ...
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1answer
30 views

Calculating quantile of distribution function

I'm having a problem with a statitics example. The calculation for my quantile is not difficult. But I'm wondering which part of my function I need to use. Let's say my distribution function is ...
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36 views

test statistics power function and size

Let a random variable X have the density p(x,θ) = θx^(θ−1),0 < x < 1 and 0, otherwise, where θ > 0. In testing H0 :θ≤1 versus H1 :θ>1,find the power function and size of the test given by C = ...
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2answers
29 views

Calculating a sample's representativeness to confirm/refute a given hypothesis?

Why hello! I'm fairly new to statistics, which is why I'm somewhat confused as to how I can approach this problem in a scientific way. The problem: Experiments are conducted to find the probabilities ...
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1answer
21 views

How do I solve Statistics Hypothesis Test and Pvalue

I was given a random sample, these are the values I got. I Need help with the Hypotheses and P value. Mean:3.1; Standard Deviation:.694; random sample:25 Perform a test of the hypotheses H0:µ=2.5 ...
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215 views

Joint probability density function using the same random variable

My original problem statement: Let $x$ be a random variable with pdf $f_x(x)$ and let y be $y=x$. (a) Find the joint pdf $f_{x,y}(x,y)$. (b) Find the conditional pdfs $f_{x|y}(x|y), f_{y|x}(y|x)$ ...
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31 views

Statistics- normal distribution

$6000$ eggs in a large store have masses which are normally distributed. The mean mass of the eggs are $42$ grams with a standard deviation of $5.5$ grams. 1) $4000$ of the $6000$ eggs have masses ...
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1answer
44 views

confidence interval population proportion question?

i think this question calls for a confidence interval calculation but im not sure. what formula do i use and how do i go about using it. using calculators i came up with an interval of 3.1. is this ...
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1answer
47 views

test statistics and power function

Let $X_1$ and $X_2$ be independent and uniformly distributed on $(\Theta , \Theta + 1)$. Consider the two tests with critical regions $C_1$ and $C_2$ given by $C_1 = \left \{ (x_1, x_2)| x_1 ≥ .95 ...
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55 views

How did they get this answer for critical value

I know how to get a critical value but I am not sure now how to do it when they added "when testing the claim that p = 1/2".... how did they get the answer for critical value ?
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1answer
39 views

test statistics

Let $X1$ and $X2$ be independent and uniformly distributed on $(0,\theta)$. Consider the two tests with critical regions $C1$ and $C2$ given by $$C1 =\{(x1,x2)\ \mid \ x1 +x2 \ge 1\}$$ $$C2 = \{(x1, ...
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79 views

Question on transformations

Two efficiency experts take independent measurements Y1 and Y2 on the length of time workers take to complete a certain task. Each measurement is assumed to have the density function given by f(y) = ...
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1answer
29 views

Identify non-interacting variables by querying a black-box function

I have a black-box function of many variables, $f(x_1, ...,x_d)$, so that I can do with $f$ is to evaluate the function at chosen points. I would like to find the maximal set of subsets of the ...
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1answer
46 views

Transformations

The length of time that a machine operates without failure is denoted by X and the length of time to repair a failure is denoted by Y. After a repair is made, the machine is assumed to operate like a ...
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18 views

Dimensionality reduction for a distribution

This question originates from a problem in high dimensional statistics. Given $n$ points in a $k$-simplex, find a map to a $d$-simplex, $d\ll k$, such that the "distance" of the points is ...
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46 views

Bias of the Kernel density estimator without bounded support of the Kernel

This is a standard result and should be easy to prove. Yet I cannot figure out the final step. Consider the Kernel density estimator: $\hat f_h=\frac{1}{nh}\sum K\left(\frac{X_i-x}{h}\right)$ ...
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65 views

A simpler proof of evaluating the limsup of a standard normal sequence

Let $\{X_n\}$ be a sequence of i.i.d. random variables following the standard normal distribution. We need to prove that $$\limsup_n\left(\frac{X_n}{\sqrt{2\log n}}\right)=1$$ This is quite easy to ...
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1answer
98 views

Interesting question regarding Weibull distribution, sample mean, and sample median

Consider a random sample from a Weibull distribution, $X_i$ ~ $WEI(1,2)$. Find approximate values a and b such that for $n=35$: (a) $P[a < \bar{X} < b]=0.95$ (b) $P[a < X_{18:35} < b] = ...
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32 views

Number of Injury claims per month

The number of injury claims per month is given by $N$ where $\\$ $ P(N=n) =\dfrac{1}{(n+1)(n+2)} $ where $0 \leq n$ Determine the probability of at least one claim during a particular month given ...
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158 views

is symmetric chi-squared distance “A” metric?

Is symmetric chi squared distance $$\int \frac{(p-q)^2}{pq}\mbox{d}\mu(x)$$ a metric? I am searching web since long time ago but I couldnt find anything. It is positive and is zero whenever $p=q$ ...
2
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1answer
69 views

Question in probability and statistics

I'm terrible at questions involving probability, so I'm hoping you can help me! Suppose there are eight people in a group, each person with a distinct first name. Each person has a card with their ...
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1answer
102 views

relative size of most factors of semiprimes, close?

when chatting about RSA a cohort just asserted something like "most prime factors of semiprimes are roughly the same size" measured in bits. ie "bits" is the number of digits in the base2 ...
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92 views

Mean and Variance of exponential function

Given this function with j and k as unknown parameters. What is the expression of Variance and Mean of this exponential function? $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 ...
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0answers
35 views

Comparison of two error distributions to determine “goodness of fit”

I am a physicist who is a few years out of doing his last course in statistics, so I am hoping to get some advice when comparing some data I recently generated. The context is as follows. I have two ...
3
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1answer
201 views

How do I 'reverse engineer' the standard deviation?

My problem is fairly concrete and direct. My company loves to do major business decisions based on many reports available on the media. These reports relates how our products are fairing in ...
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1answer
219 views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ Thereby this family contains a MLR ...
0
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1answer
69 views

Show that a statistical model belongs to exponential family

I have this statistical model: $$f_{j,k}(y)=\frac{\sqrt{j}}{\sqrt{2 \pi}}e^{\sqrt{jk}}y^{-\frac{1}{2}} \text{exp}\left( -\frac{1}{2} (j y + \frac{k}{y}) \right) \quad \quad y>0$$ In this case the ...
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3answers
257 views

Probability density function of: Z = X sin Y.

The probability density function of a random variable $X$ is a uniform distribution $U(a,b)$. Likewise, the probability density function of a random variable $Y$ is a uniform distribution $U(0, ...
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18 views

Selecting Cards

A special deck of 51 cards constists of 25 pairs and 1 wild card. The deck is distrubuted evenly between 3 players (17 cards each). What is the probability that your hand has only two pairs that is ...
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1answer
59 views

Hunting Birds probability

A hunter locates 20 geese, 25 ducks 40 eagles, 10 ostriches, and 5 flamingos. He randomly selects 6 birds to target. What is the probability at least one of each species is targeted? My reasoning $20 ...
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1answer
189 views

How to find first order density of the sample function?

I'm asked to find the first order density of the following process: $x(t)=A\cos(2\pi ft + \theta)$ where, $\theta$ is uniformly distributed over -$\pi$ and $\pi$ and f is a constant. I'm not sure ...
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1answer
83 views

vector subspaces of $(\mathbb Z/2\mathbb Z)^3$

How many possible vector subspaces of $(\mathbb Z/2\mathbb Z)^3$ are there? My idea was, to proove this as follow: $$U_b := ...
3
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1answer
48 views

Unbiased Estimator Question and Understanding

I'm having some difficulty with unbiased estimators, and wondered if anyone could help me. I believe I understand the general concepts OK, however when I come to look at some sample questions to test ...
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41 views

Variance of a difference in estimated proportions with trivariate discrete distributions

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...