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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Confusing on some concepts of sufficient principle

Im reading Chapter6 of Casella Berger's statistical inference that talks about sufficiency principle. I've been confused a lot by the definition of sufficient statistics, here it is: Basically, ...
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1answer
15 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
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1answer
12 views

Problem solving the standard deviation for a stochastic variable

Information: In a laboratory we have a vial of water that's being kept on 50 degrees Celsius. This is measured with the best thermometer in the world. The standard deviation on this thermometer is ...
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9 views

how are $t$ tables created? function relating $p$-value with $t$

I'm playing around with simulating data and comparing it to known data, and interested in how many $n$s I need in the simulated data to get significance and was wondering if there is an equation ...
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1answer
10 views

Finding the covariance of the sample variance and sample mean of a random sample with a standard normal distribution [on hold]

I am trying to work out the proof for $\operatorname{Cov}(s^2,\bar{x})$, of a random sample with $N(\mu, \sigma^2)$. I just don't know how to attack this problem and would really love some help.
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1answer
24 views

Probability and Statistics (Normal Distribution)

Having trouble with the last part of this question. Not sure how the man would divide his pile of vouchers? It seems that you could interpret this question in a lot of ways. Any tips would be ...
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10 views

Unable to evaluate Expectation in computing the CRLB

Based on the document Sec 2.5 Estimating Intrinsic dimension by Justin Eberhardt: The data points $x$ are 1 D samples of a signal obtained as the output of a linear model. It is assumed that the ...
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0answers
11 views

Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
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16 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
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3 views

Categorical Survey Heterogeneity Measure?

A survey asks the user to assign ten stick figures into four categories: strongly oppose, weakly oppose, weakly support, strongly support. The survey has been completed, so it is too late to redesign ...
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1answer
23 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
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5answers
31 views

Mutually exclusive AND independent event (help with examples) [on hold]

2 events cannot be mutually exclusive AND independent UNLESS one of the events are impossible. One example: The flipping of a coin details an independent event (flip#1 does not affect flip#2) ...
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1answer
25 views

Find the covariance of $Y_1$ and $Y_2$

I had a statistics question I was hoping for help on: Let $Y_1$ and $Y_2$ be discrete random variables with join probability function: $$f(x,y) = \begin{cases} \dfrac{y_1 + 2y_2}{18} & \text{if ...
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2answers
14 views

Can we predict next sample using the existing samples?

Suppose that I have 18 data points and I'm sampling 3 data points each time. Suppose that I have 60 samples (each has 3 data points). Can we predict the next sample (of 3 points) from the existing ...
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0answers
21 views

Central Limit Theorem proof.

I am trying to understand the proof of the Central Limit Theorem in my book. However, I don't really understand what is going on. I know the proof is assuming that the moment generating functions of ...
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22 views

Problem involving statistics [on hold]

$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt\pi}2$$ $$\explain$$
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7 views

Sufficient condition for not having infinitely small modes in a distribution

I was reading the paper Optimal Throughput and Delay in Delay-tolerant Networks with Ballistic Mobility (http://dl.acm.org/citation.cfm?id=2500432), and found the following proposition (page 305): ...
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1answer
19 views

A starting lineup consists of 2 forwards, 2 guards and 1 center. How many different starting lineups..

A certain school has $4$ forwards, $4$ guards, $3$ centers and $1$ person who can play as either a forward or a guard. How many different starting lineups can be made? I came up with 2 answers to ...
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2answers
17 views

Poisson complete statistic

I have the same question as this thread, but I cannot understand the proof. The problem is, given $f(\lambda)=\sum_{k=0}^\infty g(k)\frac{(n\lambda)^k}{k!}=0,\forall\lambda>0$. How to show ...
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1answer
14 views

geometric distrubtion and binomial confusion

I'm confused with this question - here I'm assuming our discrete random variable is $X$ which is "the number of washing machine breakdowns in a year" is this correct? If so, what is $p = 0.8$ ...
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1answer
14 views

Determining density involving scaled beta distribution

Suppose $Y \sim \mathrm{Beta}(2,1)$. If $X = \theta{Y}$ (for some $\theta > 0$) how do I determine the joint density $f(x, \theta)$? Edit: the density for $Z$ is $2z$. Would it be correct to say, ...
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0answers
16 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
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0answers
4 views

Determining the actual number of observations in a dataset

I have two datasets one is a dataset with doctors in which I have the procedures they have performed at a given hospital where the actual number of procedures is not captured by this data since it is ...
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0answers
6 views

Proof of Optimality for Approximation of Probability Spaces by PCA

I have come across a theorem that states, that the $d$-dimensional subspace found by PCA is the optimal approximation of a probability space with such a plane, in the sense that it minimises the ...
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7 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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15answers
3k views

Is the Law of Large Numbers empirically proven?

Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ...
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Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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2answers
34 views

Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
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1answer
27 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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2answers
24 views

Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
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1answer
4 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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2answers
50 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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16 views

Statistics on ordered lists

Participants of a study had to rank several proposition by their importance: 1st, 2nd, 3rd, and so on... Two cases are of interest for me: The "ideal" case: there are $n$ propositions and each ...
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1answer
17 views

Stuck on this probability question?

In a multiple choice test paper let us assume there are $m$ possible answers, if a person knows the correct answer to a question then they will mark the correct answer and that if they don't know it ...
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21 views

Finding better curved line of best fit

I have a set of hand generated data that follows somewhat closely to an exponential curve: I can come up with an exponential equation to the line that gives the values on the 3rd row, and Someone ...
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3answers
27 views

Simple probability exercise.

I have this question I need to answer but I'm a bit stuck having done no statistics before. I don't want any full solutions just hints please. A man can either drive or get the bus to his job. The ...
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1answer
28 views

Urn with marbles, unknown number of colors

When I started with this calculation I thought this was going to be a flashback from school decades ago but now after searching I'm confused if I'm over thinking it or if it's not as trivial as I ...
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0answers
26 views

How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
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22 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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2answers
68 views

Central Limit Theorem, why $n \ge 30$?

This is what I think the technical statement of CLT is: If we consider $\overline{X}_{n}$ coming from a sample of $\mathcal n$ independent and identically distributed random variables $X_{i}$ with ...
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18 views

Unable to olve the MLE: difficulty in computing the derivative

The probability that there are $k$ observations within distance $t$ of $x$ can be written as : $$\mathbb{P}[N(t,k) = k] ={n-1\choose k}[f(x)H_t]^k[1-f(x)H_t]^{n-k-1}$$ The pdf of the distance from ...
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1answer
16 views

Performing a signed test on two samples

Here is the problem My book hardly has anything about performing a sign test so I am a bit lost. Basically I know that I am testing $H_a: \mu_{13}\leq\mu_6$ vs $H_0:\mu_{13}>\mu_6$ at a level of ...
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1answer
35 views

Proving $P(A_i)=P(B_i)$ [on hold]

Suppose that $P$ is a probability on a field $F$. Consider three events $A_1,A_2,A_3 \in F$ so that $P(A_i \cap A_j) = 0$ for all $i \neq j$. Let $B_1 = A_1, B_2 = A_2 \cap A^c_1$ and $B_3 = A_3 \cap ...
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2answers
51 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
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1answer
15 views

Does the statistical frequency of patterns manipulate the probability of a given event? [on hold]

This is a question I've encountered when I first read about the Gambler’s Fallacy, I'm really wondering why it's considered fallacious? Taking statistics into consideration, If you studied the results ...
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34 views

Odds of two specific Hold'Em hands and an exact board?

The goal is to solve the probability of a Texas Holdem hand occurring where at the river one player has quad $9$s and the other player has exactly a $9$-high straight flush. (For non poker players ...
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1answer
29 views

Help reading a scatterplot

In reading the following scatterplot Would it be correct to say there is a strong positive linear relationship between the two axis? If not, could you please explain why? Also is it correct to ...
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1answer
21 views

How do I find a confidence interval of x% given an mle of θ and a standard error of sθ? [on hold]

How do I find a confidence interval of $95$% given an mle of $θ = 0.0357$ and a standard error of $𝑠_θ = 0.0057$? This is from the following question:
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1answer
12 views

How to find a confidence interval given mle and standard error?

How do I find a confidence interval of $𝑥$% given an mle of $θ$ and a standard error of $𝑠_θ$?
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10 views

Concrete example : likelihood, comparing theory and practice (genetic linkage)

Crossing tomato plants should give birth to a new population of plants of 4 types that we'll call $P_1$, $P_2$, $P_3$ and $P_4$. According to theory, the ratios of appearance of these 4 should be ...