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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Request for Statistics textbook

I am looking for a textbook on Statistical Analysis. Unfortunately most of the books I have seen, such as Statistics by DeGroot et al., are quite the opposite of the terse and lean textbooks I prefer ...
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1answer
97 views

Trick in integration with Taylor expansion

I am struggling with the expression of the LHS of the following equation. The RHS is just the Taylor expansion of the first function around point y and the differentiation wrp to the argument y. ...
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112 views

Hypothesis testing, statistics

I know you need to use hypothesis testing for this question but im not sure how to start? A manufacturer needs a part to meet detailed specifications. He will not return shipments of the part as long ...
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1answer
91 views

Confidence Intervals

How can we calculate $90\%$ confidence intervals for the variance and standard deviation of the concentration of artificial food colouring in different lots of colouring, given a random sample of $12$ ...
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1answer
201 views

statistics: probability, normal distribution

The time that customers take to complete their transaction at a money machine is a random variable with mean $\mu$ = $2$ minutes and standard deviation $\sigma$ = $0.6$ minutes. About 30% of ...
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69 views

Simple probability mass function question

I was wondering why given the probability of an event $P(X>n)=\frac{1}{(n+1)}$ then $P(X=n)=\frac{1}{n(n+1)}$. Could anyone lend me some help please? Thank you so much
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1answer
48 views

Finding expression

I don't understand this at all.. We have a Poisson distribution with parameter μ. The probability to pass the test (for the students) is p=2/3. Let X be the amount of students to pass the test. Find ...
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1answer
161 views

What is some books at the level which including this inequality and its proof?

I always wanting to looking into harder random variable/probability/stochastic process/statistics books that are harder than the intro one and have multiple random variable but easy enough to have ...
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90 views

What is the trick in the derivation? Density of a complicated function

Through one of the proofs I found a problem that really cannot solve. Imagine some density function f(x). Now, imagine that the argument is a function of the form x+c(f'(x)/f(x)). Therefore, the ...
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Probability of Obtaining the Roots in a Quadratic Equation by Throwing a Die Three Times

Question : The coefficients a,b,c of the quadratic equation $ax^2+bx+c=0$ are determined by throwing a die three times and reading off the value shown on the uppermost face of each die. i.e, ...
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52 views

Regression Proof

If the joint density function of $X$ and $Y$ is given by: $$f(x,y)= \begin{cases} 1/2, & \text{for } |x| + |y| \le 1 \\ 0, & \text{otherwise} \\ \end{cases}$$ Show that $Y$ has constant ...
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1answer
160 views

How do I show the equivalence of the two forms of the Anderson-Darling test statistic?

It's stated in many places regarding the Anderson-Darling test statistic, which is defined as $$n\int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)(1 - F(x))}dF(x)$$ that this is functionally ...
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71 views

Prediction based on posterior distribution

I am now working on this problem. Suppose a biased coin has probability $\theta$ to give heads. Suppose the prior of $\theta$ is of uniform distribution$(0,1)$. With observed information, in $n$ ...
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3answers
118 views

Marginal Density of X

I need to find the marginal density of $x$ where $$f(x, y) = xe^{-x-y}$$ Where the inputs must abide by: $x, y > 0$ My approach to solving the problem was: $\int_0^\infty \int_0^\infty ...
2
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3answers
57 views

X has pdf $f(x) = \frac{x^{2}}{18}$ for -3<x<3, what is the pdf of $X^{2}$

So this was my solution: Say, $Z = X^{2}$, then $X=\pm \sqrt{Z}$ and, $$P(Z=z)=P(X = \sqrt{z}) + P(X = -\sqrt{z}) = \frac{z}{18} + \frac{z}{18} = \frac{z}{9}$$ for $$0<z<9$$ However: ...
2
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1answer
50 views

Bayesian Problem… I think

Let X be the number of coin tosses until heads is obtained. Without knowing that the coin is fair, I assume that the probability of heads is uniformly distributed. How would I find the distribution ...
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1k views

what is the median of the CDF with the form $F(x) = 1 - e^{-(x/3)^2}$, for $x \gt 0$

Consider The distribution of lifetimes, X(in months), of a particular type of component. The Cumulative distribution function (CDF) has the form $$F(x)=\begin{cases} 1 - e^{-(x/3)^2},& \text{if} ...
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31 views

How to solve an integral like $\sum_{i=1}^n \int_{-\infty}^{+\infty} f(t) dI(y_i<t)$

how can I solve an integral of the following type: $$\sum_{i=1}^n \int_{-\infty}^{+\infty} f(t)\; \text{d}\,I(y_i<t),$$ where $I(y_i<t)=1$ if $y_i<t$ and $0$ oterhrwise. In particular I'm ...
2
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1answer
368 views

biased coin tossing where each pair of heads is separated by at least l tails.

Suppose that I have a biased coin where $\Pr[head] = p$ is very small. Let's say I tossed this coin $n$ times and I saw $h$ heads. Then what is the probability that each pair of heads is separated by ...
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2answers
457 views

Statistics: Finding posterior distribution given prior distribution & R.Vs distribution

I'm now learning Bayesian inference.This is one of the questions I'm doing. Suppose we have R.V.s $X_1,X_2,\ldots,X_n$ each have an Exponential distribution with parameter $\theta$. and prior for ...
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368 views

expectation of estimators

I have this question to answer about expectation but I only understand how to get the estimator from a table and was wondering if anyone knew the answer/knew how to explain this: Independent random ...
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1answer
85 views

Likelihood Cramér-Rao Bound.

How can I show the following necessary and sufficient condition? An unbiased estimator $ \hat{\theta} $ of $ \theta $ achieves the Cramér-Rao Lower Bound if and only if $$ \frac{\partial ...
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1answer
293 views

Expected value of $xx^{T}$ for multidimensional Gaussian

I need a bit of help understanding a step in the derivation of the expected value of $\bf{x x^{T}}$, that is, $E[\bf{x x^{T}}]$ with a Gaussian distribution. By definition, using the D-dimensional ...
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494 views

Taylor series approximation statistics

how can I show the following: Let $X_1, X_2,\ldots, X_n$ be i.i.d Poisson with mean $\lambda$. Let $Y = |\{i: X_i =0\}|$. Then $\lambda$ is estimated by $$\eta = - \log(Y/n)$$ Use Taylor series to ...
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247 views

UMVUE of λ in Exp(λ)

Suppose (X1,...,Xn) is a sample from a Exp(λ) population. Try to find an UMVUE of λ. Remember S = Sum[Xi,i=1,n] is both complete and sufficient for the Exponential Family. I'm trying to use the ...
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3answers
8k views

Standard deviation of the weighted mean [duplicate]

How do you find the standard deviation of the weighted mean? The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$ The weighted standard deviation (since it is not specified, I take ...
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427 views

Kullback-Leibner divergence of binomial distributions

Suppose $P\approx BIN(n,p)$ and $Q\approx BIN(n,q)$. The Kullback-Leibner divergence is defined by $$D_{KL}(P||Q)=\mathbb{E}(log(\frac{p(x)}{q(x)}))$$ with $p(x)$ and $q(x)$ the pdf of $P$ and $Q$ ...
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1answer
13k views

Find a Probability of a Normally Distributed Random Sample

Please help me figure out how to do this problem. I need to be able to understand how to solve problems like this. Thanks times a million! Problem: An employer is interested in the commute times for ...
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3answers
65 views

Stochastic ordering

Assume $Y$ is non negative random variable. Prove that $X+Y$ is stochastically greater than $X$ for any random variable $X$. We have to prove there that $\Pr(X+Y > x) \geq \Pr(X>x) $ for all ...
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279 views

How does standard deviation depend on data?

I've got a problem: I take a random variable, assign values to it and than I have to invent two different probability distribution functions with similar mean values (+-5%), but in the second function ...
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1answer
78 views

Necessary condition for pairwise sufficient statistic [duplicate]

I'm struggling to prove the following. If $T:\left(X,\mathbf{A}\right)\rightarrow\left(Y,\mathbf{B}\right)$ is a pairwise sufficient statistic for a set $\left\{\mu_0,\mu_1,\mu_2\right\}$ of three ...
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1answer
599 views

Rayleigh distribution

I have this question from my statistical theory course: A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
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1answer
167 views

Estimating the number of tickets bought in a lottery

A national lottery has the format where $7$ numbers are chosen from $45$ without replacement. The first $6$ numbers chosen constitute the "winning numbers", while the last number chosen is the ...
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0answers
90 views

What discrete distribution is completely determined by its mode and variance, is easy to sample, and has nice border properties?

I need to generate random ordered unranked trees that will be used to test some computer program. I'd like to incorporate some kind of control into the generation process, so that the generated trees ...
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4answers
151 views

What is a good measure of “controversy”, given a support score and opposition score?

Suppose I have a topic or discussion, and a number of "support" and "opposition" points on each side (You can also think of them as "upvotes" and "downvotes") and I want to calculate a score of how ...
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1answer
48 views

Bird's Probabilities!!!

Please, could somebody help me to figure this exercise? You collect data on the wing lengths (Yi) of 50 bald eagles in one sample. The average wing length (Y sample average) is 89 cm and the ...
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211 views

Probability Tree with three events at the same time - Help!

I'm not sure if the answers are right. I wonder if somebody could help me? Thanks Consider a population of fish (of the same species) that have different color patterns. 30% of the fish are green, ...
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1answer
325 views

What is the physical meaning of the output/ y -value of a normal distribution? (not the area under its curve)

Forgive me for my lack of knowledge regarding math terminology. I'm learning basic statistics right now, and I can see pretty intuitively that the area under a normal distribution on a certain ...
6
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1answer
134 views

Monte-Carlo for the Wasserstein metric

Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let $$ \Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
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1answer
417 views

Hypothesis Testing of the normal distribution

I have two following problems about the hypothesis testing about the normal distribution: I. A random sample of size $n$ from a normal population with unknown mean and variance is to be used to ...
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1answer
92 views

Sample size requirements in survey

If am doing some market research and want to answer the question "What percentage of the users of a service, searched for the given service online?". Lets say I go out and get people to take a survey. ...
2
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1answer
609 views

Stratified Monte Carlo

Consider the integral $I=\int_{0}^{1}e^{-x}dx$. Now consider the stratifed Monte Carlo estimate $\hat{I^{s}}$, that has $N_{st}=8$ strata. What is the variance of $\hat{I^{s}}$? What is the percent ...
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1answer
1k views

Maximum Likelihood Estimator of $f(x;\theta) = (1/\theta)x^{(1-\theta)/\theta} $

Let $f(x;\theta) = (1/\theta)x^{(1-\theta)/\theta} $$\hspace{20 mm}$ $ 0 <x <1 ,\hspace{5 mm} 0 <\theta<\infty$ I need to show that the maximum likelihood estimator of $\theta $ is ...
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1k views

How to correct/filter inaccurate speeds calculated from a GPS track

I am working on a simple application that allows the user to do a lot of different things with a GPX file (GPS track). Some things computed are: average speed, max speed, etc. The max speed values ...
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1answer
267 views

How to show that $Z = |X|$ and $-|X|$ has a a standard normal distribution? ($X \sim$ Folded standard normal distributed)

Given the probability density function of $X$ (folded standard normal distributed) is: $$f(x) = \frac{2}{\sqrt{2 \pi}} \exp\left(-\frac{x^2}{2}\right),\quad x \geqslant 0 $$ How can one show that $Z ...
3
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1answer
331 views

Number of possible positions for the Rush Hour puzzle

I'm working on a 2-d Puzzle Rush Hour which is a six * six bored that can be filled with various items : 2 blocks length horizontally vertically oriented car - let's call it 1 3 blocks length ...
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2answers
57 views

Probability help!

I am teaching a class of 100 students that has 35 men and 65 women. a.What proportion of the class are men? What proportion of the class are women? Show two different ways to calculate the ...
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2answers
2k views

What is the purpose of subtracting the mean from data when standardizing?

What is the purpose of subtracting the mean from data when standardizing? and What is the purpose of dividing by the standard deviation?
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1answer
124 views

How to compute likelihood of drawing specific set of letters from scrabble bag?

Say I have a subset of the standard scrabble tiles $B$ that contains no blank tiles s.t. $|B|=n$ Also consider I have a target set of letters $L$ s.t. $|L|=k$. These are a set of letters not tiles, ...
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82 views

Is the limit of a sequence of B-measurable functions itself B-measurable?

Let $\left(\Omega,\mathcal{A}\right)$ be a measurable space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Let $g,f_1,f_2, f_3,\dots$ be real-valued functions with domain $\Omega$ ...