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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Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
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43 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
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Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
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19 views

probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
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Which distribution would be the most appropriate?

What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns? e) The size of an automobile insurance claim I'm thinking that the distribution ...
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26 views

An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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24 views

Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
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21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
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22 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
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50 views

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$. Then which of the following is true? The standard deviation of $X$ is smaller than that of $Y$. The mean of $X$ is smaller than that of $Y$. The ...
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10 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
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28 views

Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
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How to find expectation of birth-death process [on hold]

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?
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Proof of the DKW inequality

My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= ...
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1answer
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Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
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In the context of ordered statistics, each of Y(1),Y(2),…,Y(n) a single observation or distributions that are I.I.D?

In statistics one aspect of the I.I.D. concept that bothers is when I think about it in the context of ordered statistics. As most of you already know, $Y_1,Y_2,Y_3,...,Y_n$ are I.I.D. when the ...
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Please help me create a mathematical model that can be used to compare two different strategies [closed]

I have been pondering this question: "Is it better to be balanced, or focus on one area?" Let's say that I can decide how to divide my investment between two products. I want to know if it is better ...
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31 views

statistics help [closed]

A charity solicits donations by phone. From long experience the charity’s director reports that 60 percent of the calls will result in refusal to donate, 30 percent will request more information via ...
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If $P(A) = 0.2$, $P(A \cap B) = 0.1$, $P[(A \cup B)'] = 0.3$, what is $P[(A \cap B) \mid (A \cup B)']$?

Suppose events $A$ and $B$ are such that $P(A \cap B)= 0.1$ and $P[(A \cup B)'] = 0.3$. If $P(A)=0.2$, what is $P[(A \cap B) \mid (A \cup B)']$? I tried solving it by using the conditional ...
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46 views

Covariance/Correlation Proof

I'm having a little problem with a statistics problem I am working on. I'm not really sure where to start to prove the two statements. Any help would be greatly appreciated. Let $x$ and $y$ be ...
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Notation Bayesian Statistics $\propto$

I often read the following notation: $\propto$. How is this sign called and what is the definition of it?
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How to analyse data set

I have a data set from work I'm not sure how to analyse. We have a bunch of people we've contacted (about $1,000$) at a random time during a project they're conducting. That time is expressed as a ...
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1answer
46 views

The mathematics of Correlation is not equal to Causation

In statistics, it is a common practice to say that "correlation does not mean causation", and mostly the proof for this is given by examples. While that is good for the intuition, it's not rigorous. ...
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Proposal Criteria Analysis

We've just finished evaluating three proposals on the following criteria: Criterion Point A1 125 A2 125 A3 100 A4 150 Cost 500 ----- ----- Total ...
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4 views

Sufficient statistics 2

Given $X_1, ..., X_n \ iid$ from a family of distributions parametrised with $\theta$, $Y = g(\underline{X})$ sufficient statistic for $\theta$, $Z = h(\underline{X})$, that $Y$ and $Z$ are ...
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1answer
28 views

Analysis of calls to a call center using poisson distribution

I have a set of data from my workplace where we note how many support calls we receive. I have been playing around with it in my spare time just to see if I could predict anything interesting. I have ...
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21 views

Moment Estimator

I have been given Independent sample variables $X_1,...X_n$ that have a common p.d.f $$f(x,\theta)=\frac{10\theta^2}{x^2},$$ where $0<\theta<x$. How do I go about in finding the moment ...
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Statistics - Mean of Ungrouped Data

Two weeks before James opened technology titans he launched his company web site. During those 14 days James had an average of 24.5 hits on his Web site per day. In the first two days that technology ...
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29 views

Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
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13 views

proof for the equivariance of the MLE

I am self-learning statistics and I read about the theorem that under some conditions the MLE is equivariance. I couldn't find any proof for that theorem. What are the conditions and what is the ...
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32 views

What does the capital $E$ notation followed by curly bracket mean?

While reading through a statistics book earlier today I came across a notation I'm unfamiliar with and can't find a way to search for it. It is not expected value $E[\,]$, but instead the following. ...
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1answer
27 views

Why was poisson distribution introduced?

I am studying probabilites and the notion of poisson random variable was introduced in the class. But it seems to me that the introduction of poisson random variable is to provide a easy approximation ...
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35 views

Finding MLE for $\mu^{2}$

The problem says the following: Let $X = (X_{1}, ..., X_{n})$ be a random sample, where $X_{i} \sim N(\mu_{0},1)$, where $\mu_{0} \in \mathbb{R}$ is unknown. I do not have problems calculating the ...
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1answer
43 views

Obtaining quadratic equation using Least Squares Method

This question is most likely extremely trivial, but I'm having some difficulty obtaining the least squares equation from the following data points: {{1.08, 0}, {1.07, 0.0659232}, {0.97, 0.1695168}, ...
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8 views

U-statistics and Independent Sum

I have i.i.d. paired samples (X,Y): $(X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)$ I compute the statistics $\sum_{i \neq j} X_i \cdot Y_j$ People have told me that the above is actually a sum of $n$ ...
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48 views

Finding the distribution of $n$ times the minimum of $n$ exponential random variables.

I'm having trouble with this question: Let $X = (X_{1}, \ldots, X_{n})$ be a random sample, where each $X_{i}$ is an exponential random variable with mean $\lambda_{0} \in (0,\infty)$, which is ...
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Math Test review on probability and statistics

A committee of 11 members is voting on a proposal. Each member casts a YES or NO vote. On a random voting basis, what is the probability that the proposal wins by a vote of 8 to 3?
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Statistics: Adding up averages

Let's assume we the average income for a journalist is $10000$ (in the whole country). In a certain state X the average income for this profession is $10700$, therefore 7% higher. Also, the average ...
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Bound of diameter in Erdos-Renyi

I would like to compute the bound of the diameter in random graph $G(N,p)$ following Erdos-Renyi model. Anyone can tell me how to compute this bound? Thank you so much.
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Fitting a model to a collection of binomial proportions, based on varying (large) sample sizes.

I have a multi-parameter bivariate function, say $f(i,j)$ that I want to use to predict the entries of a matrix $M(i,j)$, the entries of which are binomial probabilities based on varying sample sizes, ...
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1answer
9 views

linear transformation of factorial moment generating function

Let $X$ be a discrete random variable with factorial moment generating function $\psi_X(t)$ and define $Y=aX+b$, where $a$ and $b$ are constants. Express the factorial moment generating ...
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Probability of reception from n users

Say I have 4 boxes on the ground. At every time interval the nth user(n = 1,2,3...) has some probability of throwing 0, 1, 2, 3, or 4 balls into the boxes. Each box can only hold one ball, and the ...
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Help with Bionomial Distribution Question part e) [closed]

Q2 ABC Airlines’ first class cabins have 10 seats in each plane. ABC’s overbooking policy is to sell up to 11 first class tickets, since cancellations and no-shows are always possible (and indeed are ...
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1answer
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Statistics - Probability

QUESTION If a new type of torch battery has a voltage that is outside certain limits, that battery is characterised as a failure (F); if the battery has a voltage within the prescribed limits, it is ...
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1answer
33 views

Ratna sagar textbook of 9th class maths [closed]

The question is: The mean of 15 numbers is 25. If 4 is subtracted from every number, what will be the new mean?
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Degree of nodes in Erdos Renyi model

I see in the E-R model with a random graph $G(N,p)$ on $N$ nodes and probability of edge existence $p$, the probability that a node has degree $d$ is $$ P(d)=\binom{N-1}{d} p^d (1-p)^{N-1-d}$$ Give a ...
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18 views

How to perform statistical test for two sets of points?

(I have asked this question originally on Cross Validated; however, no good answer and someone suggested me to ask the question here). Thanks a lot in advance if anyone can help. We know that we can ...
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1answer
34 views

Testing of hypothesis

Following is a question from my textbook. My approach is different from one explained in the book. I cannot understand what is wrong with my solution. I have explained both solutions below. Kindly ...