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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confused about meaning of a expectation of a function

https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff#Derivation well,in the "Derivation" part of the wiki link. i don't figure out why $E(f)=f$, does it imply that the function $f$ is ...
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4answers
40 views

Statistics: Conditional Probability

$P(A│B)=\frac25$ ,$P(B)=\frac14$, $P(A)=\frac13$. Find $P(A\land B)$ $P(B|A)$ Here is what I did: Part 1. $$P(A\land B) = P(A) \cdot P(B)\\ = \frac13\cdot\frac14=\frac{1}{12}$$ Part 2. ...
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3answers
14 views

Conditional Probability

1) A card is drawn from a standard deck of $52$ cards. The card is drawn from the pack and not replaced. A second card is then drawn from the deck. Determine the probability: $a)$ that the ...
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0answers
18 views

How could an estimator be biased but consistent according to mathematical definition?

According to the definition, an estimator can be biased, if $E_{\theta}[\hat{\theta}]\ne\theta$, with $\theta$ as parameter for a distribution we want to get from samples. While the estimator can be ...
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0answers
17 views

Does this resilience/resource scheduling analogy make sense?

A firend has recently presented an analogy for the rescheduling of Doctors (big topic in the UK atm) across a 7 day week as opposed to a 5 day week with a skelton staff at weekends - I'm ignoring ...
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1answer
28 views

Find the joint probability density given the support set

Suppose that the support set of $(X,Y)$ is $$S_{X,Y}=\{(x,y)\in\mathbb{R}^2: x \geq 0 \text{ and } 0 \leq y \leq e^{-x/3}\}$$ $(X,Y)$ is uniformly distributed on $S_{X,Y}$. a) Find the joint ...
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2answers
20 views

How can i calculate the result from this formula?

There is a question posted here that the "Generalized Inclusion-Exclusion Principle" is used to be answered. See here: Probability of number of unique numbers in $37$ Roulette Wheel spins. In this ...
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2answers
38 views

Proving a Trick to More Quickly Calculate N-Step Transition Probabilities

So, I have been working on a homework problem all day that asks me to prove that: $P^n= \Pi +Q^n$ where P is the transition matrix of a finite-state regular Markov Chain, $\Pi$ is a matrix whose rows ...
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1answer
16 views

Why is $Y$ and linear function of $X$ if the correlation equals $+1$ or $-1$?

Just looking for a proof of $\operatorname{Cor}(X,Y) = \begin{cases} +1 & \text{if } a>0, \\ -1 & \text{if } a<0, \end{cases}$ where $X$ and $Y$ are random variables such that $Y=aX+b$ ...
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16 views

What is the saddle-point approximation? [on hold]

I want to take advice which books are useful to understand saddle point approximation. Can you give suggestion about that ? Also, if you explain what is the saddle point approximation, I will be so ...
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2answers
15 views

Finding of $\hat{\theta}_{MLE}$ of $f(x; \theta) = (\theta + 1)x^\theta$

Let $X_1, \cdots, X_n$ be a random sample from the PDF: $f(x;\theta) = (\theta + 1) x^{\theta}$ with $0<x<1$ and $\theta > -1$. The likelihood function is: \begin{align} L(\theta) &= ...
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1answer
32 views

Convergence in Distribution to the normal distribution.

let $ X_1,X_2+,...,$ be independent and identically distributed random variables with Poisson Distribution, does $$ \frac{1}{\sqrt{n}}\sum_{i=1}^n(X_{2i-1} - X_{2i})$$ Converge in distribution to ...
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1answer
36 views

Find 99% confidence interval

Can anyone help in these questions? The Government is interested in knowing the average gas consumption of a typical family. One hundred houses are chosen and asked to measure the gas consumption ...
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0answers
14 views

Central limit theorem for uncorrelated identically distributed random variable

I have a sum of random variables as bellow $$Y=\sum_n A_n=\sum_n B_n\times C_n$$ where $B_n$s are correlated Gaussian random variables with zero mean, variance $1$ and correlation ...
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0answers
14 views

Is truncated normal CDF a decreasing function of $\mu$?

Suppose $F(x;\mu,\sigma^2,0,1)$ is the CDF of a $N(\mu,\sigma^2)$ random variable truncated on the unit interval $(0,1)$. I'd like to show that $\frac{\partial}{\partial \mu} F(x;\mu,\sigma^2,0,1) ...
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0answers
8 views

If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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0answers
18 views

In a sweepstakes giveaway scenario, how does having 2 chances to win the same prize affect the overall odds?

In a sweepstakes giveaway scenario where total entries are expected to result in final odds of 1:93,150.685 for/against a single entrant (after adjusting for multiple entries) and can be won by either ...
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0answers
12 views

Maximising returns Limiting Risk

This is probably a simple question/solution, but I'm no math expert. I'm looking into a Facebook group that provides bet to try and get up to 50k, you may have heard of it. The premise being that you ...
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4answers
25 views

Two definitions for standard deviation

There seem to be two canonical definitions for the standard deviation. $$\sigma_n = \langle(n - \langle n\rangle)^2\rangle^{1/2}$$ and $$\sigma_n = \sqrt{\langle n^2\rangle - \langle n\rangle^2}$$ ...
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17 views

Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence? Referring to references is appreciated.
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1answer
20 views

Jensen's inequality in derivation of EM algorithm

I am going through the derivation of EM algorithm and got stuck on understanding the following steps: Notes showing EM algortithm derivation For the equality to hold, f(x) has to be an affine ...
3
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0answers
39 views

Concentration inequality for sum of squares of i.i.d. sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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1answer
20 views

How to Graph Relationship Between 3 variables [on hold]

I have 3 variables that I'm interested in understanding the relationship between. ...
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0answers
10 views

Constructing a copula that satisfies the desired condition

Exercise 2.8 in Roger Nelson's An Introduction to Copulas asks the reader to construct a copula $C(u,v)$ not equal to $\max(u + v -1 , 0)$ that satisfies the property $$ C(u,u) = \max(2u - 1,0) $$ ...
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1answer
32 views

Probability that first and last are same gender?

There are twelve members on a team, $8$ women and $4$ men. Out of these $12$, a $4$-person relay team is being chosen. What is the probability that the first and last leg of the race are the same ...
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0answers
42 views

Sampling bias using addresses as frame

A selected address in the sample has no residents. Will this cause bias in the results of a survey where house addresses are used as the frame? I think it wouldn't, because an address with no ...
2
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1answer
16 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
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1answer
37 views

Probability of Type II error

I have these values: A sample of size 100 is taken from the population. Standard Deviation is 5 Mean 125 H0: $ \mu=$ 125 against Ha:$ \mu<125 $ Accept H0 if the sample mean ...
0
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1answer
17 views

Calculating 5 different ranges for people resource management

I am working on a project for my company. My team is building a project charter template. In this template needs to be a drop down that estimates how many full-time employee days(FTE) will be ...
2
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1answer
32 views

Is multiplication of a correlated random variable and a independent random variable, an independent random variable [on hold]

I have a random variable that is a multiplication of two random variables as bellow: $$A_n=B_n\times C_n$$ $B_n$s are identically distributed with zero mean and are correlated for different $n$s and ...
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1answer
29 views

Probability Type I error

I have these values: A sample of size $100$ is taken from the population. Standard Deviation is $5$. Average is $125$. $H_0$: $\mu =125$ Against $H_a$: $\mu <125$ Accept $H_0$ if the sample ...
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2answers
45 views

drawing cards in the deck [on hold]

Suppose 3 cards are drawn from a shuffled 52 card deck. The face cards are the Jacks, Queens, and Kings. Let A = {all diamonds} and B = {All face cards} Are the events A and B independent? ...
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0answers
10 views

Error in Six Sigma Rolled Throughput Yield Calculation

I've been looking at running some calculations on process yields and landed on Rolled Throughput Yield and Six Sigma. I had been using the product of all process yields in the past, no idea Six ...
4
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1answer
53 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
0
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1answer
25 views

finding UMVUE $P ( X_1> t)$ [on hold]

Suppose $ X_1,\ldots,X_n$ are i.i.d. random variables with density: $$f(x_i;\theta)=\theta x_i^{-2}$$ $$x_i>\theta$$$$\theta>0$$ The smallest order statistic $X_{(1)}$ is sufficient and ...
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1answer
21 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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1answer
31 views

Confused with the power set of an integer

I am going through The Maximum Degree of a Random Graph by RIORDAN et al. On the second page, the notation $\mathbb{P}(\mathcal{D})$ is used which I assume the power set of the set $\mathcal{D}$. ...
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0answers
15 views

Deriving sample size using Hoeffding's Inequality

I want to use Hoeffding's Inequality to determine the necessary sample size $n$ to construct a confidence interval of $\epsilon$ and $\alpha$. I've consulted the Wikipedia article and am confused as ...
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0answers
11 views

How do you derive the confidence interval of linear fit parameters?

I have read a few ways of deriving the expressions for the parameters of a linear fit (i.e. slope and intercept) for a given set of values $X$ and $Y$. However, I have not found a treatment of: The ...
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0answers
16 views

How to determine the correct $\epsilon$ value for comparison of real values of a custom data set?

When comparing two real values we usually define an $\epsilon$ value and say they are equal if $|a-b| < \epsilon$. I evaluate Pareto approximations using different performance metrics also known as ...
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1answer
38 views

How come this Poisson formula equals 1

In Poisson Random Variable: $$\sum_{x=1}^\infty \frac{e^{-\lambda}\lambda^{x-1}}{(x-1)!}=1$$ Why does this equal $1$? What property is this?
2
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1answer
79 views

Coupon Collectors Problem with Packets: Clarifying Wikipedia

The Coupon Collector's Problem (CCP) is very useful in many applications. However, the "default" CCP is relatively simple: suppose you have an urn containing $n$ pairwise different balls. Now you want ...
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0answers
14 views

SQL joins and analysis

Say we have a users table and an events table and what sort of analysis can be done? Also, what is some SQL statements to describe the analysis of these 2 tables?
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3answers
31 views

How does one find the density of the $k$th ordered statistic?

Let $X_1,\ldots,X_n$ be $n$ iid random variables. Suppose they are arranged in increasing order $$X_{(1)}\leq\cdots\leq X_{(n)}$$ The first ordered statistic is always the minimum of the sample ...
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1answer
18 views

Proving Weak Law of large numbers by Markov's inequality

Hi I am trying to solve the problem 5.13 of the book Statistical inference by George Casella and Roger L. Berger. The problem is Formulate and prove a version of the WLLN with a weaker assumption ...
2
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1answer
40 views

Probability of a random Permutation [on hold]

Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last ...
2
votes
1answer
34 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
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2answers
32 views

How do you interpret conditional probability when two events are switched?

Before I pose my question, I want to emphasize that I am not seeking a homework help or steps on how to derive the answer, for I already know the solution, and how to get it. What I am seeking is, how ...
2
votes
1answer
51 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
3
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1answer
26 views

Urn probability replacement problem

An urn contains $10$ red and $10$ white balls. They are taken out at random one at a time. Find the probability that the fourth white ball is the fourth, fifth, sixth or seventh ball drawn if the ...