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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Correct Way to Break Up Range Into N Approximately Equal Discrete Bins

This seems simple, but none of the solutions I've tried work well. For example, for a range [0, 67), how can you split it up into 20 approximately equal sized discrete bins without introducing ...
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13 views

Auto-correlation calculation

Is auto correlation of lag $k$ the same as auto correlation of order $k$? If not, how do their formulas differ? For lag $k$ is the denominator: $\sum_1^n(x_t-\bar{x})^2$. For order $k$ is the ...
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6 views

Statistics, Chapter random variables and discrete probability distributions [on hold]

An elementary school teacher learned that 35 percent of school age children have at least 3 cavities. The teacher has 30 students in his class. What is the standard deviation? (Round your answer(s) to ...
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19 views

Var(Y_{i}) = σ^{2})

In the system regresion the supose that: 1) $E(\epsilon_{i}) = 0$ 2) $Var(\epsilon_{i}) = \sigma^{2}$ $E(Y_{i})=\beta_{0}+\beta_{1}\times X_{i1} + \ldots + \beta_{p}\times X_{ip}$ How I prove that ...
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2answers
32 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
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1answer
28 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
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26 views

What is the variance E[A]^2, statistics? [on hold]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
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2answers
17 views

Probability Distribution Function?

An urn contains 8 green balls and 17 yellow balls. A ball is drawn from the urn and its color is noted and then the ball is placed back in the urn. 5 balls are drawn this way. Let $X$ denote the ...
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1answer
32 views

How to compute the question [on hold]

The senior sales agent for Regional Real Estate Company closes a deal 71.5% of the time. a) If the manager believes that the agent will see 14 prospective buyers next week, what is the probability ...
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1answer
21 views

Likelyhood of Poisson Distribution

The number of accidents in a week follows a poisson distribution with mean $\lambda$. Likelyhood is given as $$L(\lambda)=\frac{ \lambda^{\sum_1^n x_i } e^{-n\lambda}} { \prod x_i!}$$ However only ...
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0answers
13 views

Predictive analysis based on history

Let me first say that I am a CS person and my knowledge about statistics is quite basic. I am trying to see what predictive analysis to use for a problem I am trying to solve. I will try to make my ...
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0answers
24 views

Predicting profit with price variation

I am currently working on a high school project that aims to predict profit from X amount of items to Y amount of profit based off a deviated sale price. For instance: I sale 10 cookies for 10 ...
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22 views

Normal probability distrubution.

any student who scores in the top 4% of students on the SMRT standardized exam. SMRT scores have a mean of 260 and a standard deviation of 22. What score does a student need to attain in order to ...
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16 views

Second derivatives of MLE to multivariate normal distribution

I want to calculate the second derivatives to the MLE's $\hat{\mu}$ and $\hat{\Sigma}$ to confirm that the extremums indeed are maximums to the multivariate normal distribution $$ ...
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1answer
26 views

Probability that an event will occur X times in a row at any point in Y trials?

Event AA has a $60\text{%}$ failure rate. Given $256$ trials, what is the probability that at some point event AA will fail $9$ times in a row? Is there a formula that is fairly plug-and-play? I ...
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1answer
20 views

How to find the PDF from the CDF?

I have the following CDF: F(x)= \begin{cases} 0 & \text{if } x < 4 \\ \frac{x-4}{4} & \text{if } 4 \le x < 8 \\ 1 & \text{if } x \ge 8 \end{cases} I know that the PDF is equal to ...
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3answers
71 views

Show that $P(X=c)=1 $for some constant c

Suppose $X$ and $Y$ are independent random variables, also $X$ and $X-Y$ are independent. Prove that $$P(X=c)=1$$ for some constant c. I tried using moment generating function, please give me some ...
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1answer
24 views

A joint pdf question [on hold]

I need help over a question. I appreciate all helps.Thank you.
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1answer
13 views

Sampling distributions question

LEt x bar be the mean of a random sample of size n=48 from the uniform distribution on the interval (0,2); that is, f(x) =1/2 for 0 for this problem i am confused because don't we need to know mean ...
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23 views

In 60,000 CABG surgeries, what is the expected number of retained foreign bodies? I cant figure out where I am going wrong on this.

In CABG surgery, there is a 0.00015 probability of a retained foreign body (e.g., a sponge or a surgical instrument) left inside the patient. (a) In 60,000 CABG surgeries, what is the expected ...
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17 views

In Northern Yellowstone Lake, earthquakes occur at a mean rate of 1.3 quakes per year. Let X be the number of quakes in a given year. [on hold]

In Northern Yellowstone Lake, earthquakes occur at a mean rate of 1.3 quakes per year. Let X be the number of quakes in a given year. (a) Justify the use of the Poisson model. ...
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20 views

recognize the distribution corresponding to this characteristic function

The characteristic function of a random variable X is given as $$\frac{3+cos(t)+cos(2t)}{5} $$, what is the distribution of X? I was thinking of the discrete random variable X=,0,1,2 with mass ...
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1answer
22 views

What is the probability that all are eligible? What is the probability that at least one is ineligible? [on hold]

Past insurance company audits have found that 2 percent of dependents claimed on an employee’s health insurance actually are ineligible for health benefits. An auditor examines a random sample of 10 ...
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Uni-variate Moving Average Theta coefficients

Consider the Uni-variate Moving Average Models (MA models) MA(1) $$x_t = \mu + w_t +\theta_1w_{t-1}$$ or the second order moving average MA(2) $$x_t = \mu + w_t +\theta_1w_{t-1}+\theta_2w_{t-2}$$ ...
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6 views

Multiplying/dividing mean and std dev of two different values

I'm working on predicting outcomes of games given statistics from the two teams involved. Two of the statistics I have are below: AGA (average goals against): The arithmetic mean of number of goals a ...
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15 views

Contingency table with a 0 value in a concrete problem.

I don't know how to deal with this problem: In order to evaluate the relationship with a risk factor and a disease we have the following case-control study: \begin{array}{|c|c|c|} \hline Risk Factor ...
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2answers
22 views

What is the expected value of the mean of the highest $m$ numbers in a population of $N$ normally distributed random variables?

Suppose that I randomly generate $N$ numbers according to the standard normal distribution, $\mathcal{N}(0,1)$. Then suppose I pick the highest $m$ numbers, $x_1\leq x_2 \leq \cdots \leq x_m$. ...
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1answer
21 views

Could not find right formula for Conditional Probability [closed]

P(A) = 0.20 P(B) = 0.62 Assuming that A and B and mutually exclusive, the conditional probability $P(A|B^c)$ is equal to... A) 0.769 B) 0.797 C) 0.948 D) 0.526 I've tried all the formula but I ...
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43 views

Mutual Exclusivity

Suppose that a study is being done on all families with one, two, or three children. Let the outcomes of the study be the genders of the children in descending order of their age. A. List sample ...
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28 views

A linear algebra textbook that is advanced enough as a prerequisite to read time series and econometric textbook?

A linear algebra textbook that is advanced and comprehensive enough as a prerequisite to read time series by Hamiliton and econometric by Hayashi? If possible, please also answer on which statistics ...
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1answer
29 views

Why does β=α give a symmetric standard beta pdf?

I know that β=α is what will give a symmetric standard beta pdf, but why is this so?
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24 views

Negative Correlation and Securities?

The following is a problem in the textbook I am trying to do as I practice for an exam. I would appreciate some help on this question I have completed part a. I have determined expected return in ...
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14 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
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2answers
23 views

Covariance for equal variances

If X and Y are independent random variables with equal variances, find Cov(X+Y, X-Y). I am confused on how to do this? I feel like I am over thinking this question.
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orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
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13 views

normal distribution

An urn contains 8 green balls and 17 yellow balls. A ball is drawn from the urn, its colour is noted, and then the ball is placed back in the urn; 15 balls are drawn in this fashion. (a) Would it be ...
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2answers
11 views

Variance & Joint Density Function

$X$ and $Y$ have joint density given by $$f_{XY}(x,y)=\begin{cases}2,& 0≤x≤y≤1 \\0,& \text{elsewhere}\end{cases}$$ a) Find $\text{Var}(Y|X=x_0)$. b) What is the answer if $x_0$ is not in the ...
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1answer
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approximations of discrete distributions

if X is b(100,0.1), find the approximate value of P(12<=X<=14),using a) The normal approximation b) The Poisson approximation c) The binominal using the binominal approximation I think it ...
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3answers
37 views

P(A|B^c) given the following

Given $P(A) = 0.2, P(B) = 0.6$, where A and B are mutually exclusive, find the conditional probability $P(A|B^c)$. How do I determine this answer? I've been trying to figure it out for hours.
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Best Estimate for Mean Difference

To compare two methods for the determination of uric acid, 23 blood samples from 23 individuals were divided and analyzed both by the colorimetric method and the uricase method. Suppose the sample ...
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17 views

What makes standard deviation a good measure of dispersion from the mean?

Standard deviation formula: If I just want to measure the amount of dispersion from the mean, why can't it be just an average of the absolute differences of each term from the mean? Something like ...
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1answer
20 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
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8 views

How to calculate probabilities of win, draw and loss based on the ELO system

Consider one game between two opponents (I am actually thinking about chess but it doesn't have to be that). The ELO rating system gives the expected score $E_A$ and $E_B$ for both players. That is ...
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10 views

Negative binomial distribution's mgf!

I've got problem. during my trying to find mgf(moment generating function) of negative binomial distribution, I found myself cannot understanding process. How does second line become third line? ...
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1answer
25 views

Infinite boundary for random variables

I have a question Suppose that X and Y are random variables with joint pdf is given by and zero otherwise. I need to find marginal and conditional pdf's.But I don't know how to intagrate over an ...
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1answer
48 views

Probability question - statistics

 Could somebody please help me, I have been trying to calculate this problem all day, but with no success. Here is the problem: "A teacher was asked by her principal to select 7 students at random ...
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1answer
38 views

95 percent of satellites launched are successful. What's the probability that,the next four launches, there will be no mishaps and exactly one mishap? [closed]

I honestly have no clue how to do this. Any help would be appreciated. I have tried everything.
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1answer
77 views

A baseball player's batting average is $0.31$. If, in a given game, [closed]

he bats four times, what is the probability that he will get: no hits? at most two hits? at least two hits?
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38 views

If $Y=X\beta+\epsilon$, prove that the least square estimator $\hat\beta$ is independent of $Y-X\hat{\beta}$

Let $Y=X\beta+\epsilon$, where $Y$ is an $n$ by $1$ vector, $X$ is an $n$ by $p$ matrix with full rank and $\epsilon$ is an $n$ by 1 vector of random errors independently and normally distribution ...
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0answers
59 views

Probability question in statistics [closed]

Suppose I give you a list of 24 problems to study, from which I will randomly pick 14 questions for your first midterm exam. For whatever reason, you prepare for the midterm exam by completing and ...