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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conditional probability - finding a single event

Context: P(L) = 0.1, P(M | L) = 0.45, P(M | L') = 0.51 M = Male M' = Female L = Left handed L' = Right handed 1) Find P(M): So I knew I had P(M | L) and P(M | L') so I tried to add the two as it ...
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23 views

Coefficient of variation - Help me decode this text

Doing some exercises I hit upon this wall of text: Coefficient of Variation should only be computed on ratio scales (i.e., data where there is a "true" zero, like temperatures in Kelvin, or ...
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1answer
21 views

Standard Deviation and Mean

take a look at this question: The sample from Population A has a mean of 35 and a standard deviation of 1. The sample from Population B has a mean of 45 and a standard deviation of 15. Which of ...
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25 views

Randomized Block Design for repeated trials - Please help

Suppose I measure reaction time of 4 students, initially before any treatment, then after treatment 1, and finally after treatment 2. (Note all treatments will eventually be applied to all students) ...
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1answer
25 views

Question on finding the Jacobian

I have a question which is as follows: The random variable U has the pdf $n \dfrac{u^{n-1}}{\theta^{n}}$ , $ 0 \le u \le \theta$ This is not independent of the parameter θ. Let Y=U/θ. Use the ...
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6 views

some knowledge about comare means of two group

I'm studying a bout statistic. I can compute very well. But i don't really know well. I want to know clearly about theory. And i have this problem: ...
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15 views

General procedure Vs pooled variance

When we compare means of two group. With independent sample, we have two method, but i don't know when use choose which one and when choose the other. Please help me classify them Two method are: ...
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2answers
19 views

stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
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How to Calculate $\sigma_x^-$

I have this question and I need to calculate $\sigma_x^-$ first but I'm not sure how A confidence interval is desired for the true average load loss $\mu$ (watts) for a certain type of induction ...
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1answer
23 views

Prove that $COV(h(x),g(x)) \leq 0$ means the different direction for $h,g$

(Covariance Inequality) Prove that if $g$ is nondecreasing and $h$ nonincreasing, then $$ E(g(X)h(X)) \leq E(g(X)) E(h(X)) $$ I know that it is equivallent to prove $COV(g(X),h(X)) \leq 0$ if $h$ ...
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14 views

Statistic question [closed]

Hello and thanks for looking at my question. I'm having trouble finding an explanation of how to do this problem. Any help would be greatly appreciated The length of a human pregnancies are ...
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1answer
25 views

Probability: 30 balls in a bucket, homework

i need some help with some homework, first time i am doing probability and statistics, id like to know if my 2 answers below are correct, and how i can solve the remaining 2. There are 30 balls in a ...
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1answer
29 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
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14 views

Statistics questions

Hello and thanks for looking at my Question. I am having trouble figuring out how to answers these two questions. I cant find a good explanation in the textbook I have so I have come here for help. ...
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12 views

Help with The Nature and Perception of Fluctuations in Human Musical Rhythms paper.

I'm a electronic musician. I've found this article about human rhythm. I know some electronic musicians are already using it to humanize electronic drums (e.g. James Holden) but I'm having trouble ...
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25 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
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34 views

Does vigorous exercise affect concentration?

In general, the time needed for people to complete a certain paper and pencil maze follows a normal distribution, with a mean of 30 seconds and a standard deviation of three seconds. You wish to see ...
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32 views

What is this pattern?

I have a pattern of numbers from some simulation that I'm trying to piece backwards into some sort of formula. All I can see right now is that they maybe have to do with integrals? The numbers (which ...
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5 views

The confidence of a transforming function

Let's say we have a set $I$ of items. We have $4$ functions $a$, $b$, $c$, and $d$ assigning a number to each of the items in $I$ ($a,b,c,d : X \rightarrow \mathbb{R}$). Further, we have $2$ functions ...
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33 views

Combining confidence intervals for sums of generic random variables

So my fiancee is a civil servant and asked me for help with the following problem. She has been given a collection of upper and lower bounds on expenditure for a collection of projects like: ...
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1answer
10 views

How to choice between paired sample and Independent sample

I don't know how to decide to choose between pair and Independent to compare mean of two groups. Here is the excersise from a book: ...
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1answer
7 views

Fishers Exact test alternative to chi-squared?

Background*strong text* Am analysing results from a questionnaire (n=130). The respondents fall into four categories (A / B / C / D), and each question/statement has 5 possible answers (Likert Scale ...
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1answer
45 views

Find an unbiased estimator

Let $X$ be an r.v defined by $P(X=0)=p$ and $P(X=1)=1-p$. Find an unbiased estimator for $2p$. My solution: $E(X)=1-p$ so $2-2E(X)$ is unbiased. Is this correct?
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1answer
28 views

Expectation by conditioning

Problem: Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. What is the mean number of necessary trials? Start on solution. ...
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1answer
29 views

Confidence interval for n=1, unknown standard deviation

Is there any way to calculate a confidence interval (or otherwise gauge the reliability of a sample) when you have a sample size of one and you don't know the population standard deviation? I work at ...
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34 views

Meaning of covariance

Can someone please give me an intuitive explanation for the meaning of covariance between two random variables? What does it measure?!
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Markov chains - Proof of how to check recurrent states

Question 1 I read a proof of how to check recurrent states. There is one = sign that I do not understand, see the image. ...
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44 views

Almost sure convergence of a sum of independent exponential random variables?

I'm in difficult with this exercise... I hope someone can help me. Let $X_1,X_2,...$ be independent random variables, $X_n\sim \exp(\lambda_n)$, where $$0 < \lambda_n\rightarrow \lambda , \lambda ...
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1answer
26 views

Null Hypothesis and Confidence Intervals

I was told that If I reject the null hypothesis than my CI will not include the population mean. However the problem that I just did has a population mean of 2.72 and a lower CI of 2.5 and an upper CI ...
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1answer
23 views

Upper and Lower Bounds for Confidence Intervals on a one tail hypothesis test.

CI(99) = xbar ± Zcv( σ / √ n -1 ) My question is if the test is a one tail hypothesis test, do I still do two separate equations + and -. Here is the formula filled in (if it is the correct ...
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1answer
13 views

Why am I getting 84.25 instead of 97.68 for my Confidence Interval?

Why am I getting 84.25. I am doing 97.96*(6/sqrt[50-1]) = 97.96*(6/7) I am obviously not doing this correctly please help. It has been sometime since I have done any math.
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What is this probability that this event could occur?

Records show that 30% of the people, listed as passing the Firefighter's exam in Denver, are republican; the remainder as democrats. Last year, 30 people were hired as firefighters for the city. (25 ...
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1answer
13 views

statistical significance in non normal distributed data

I have randomized strings undergoing two different tests. There's a pretty clear difference in their proportions, but it reminded me of apstats where you would often do 2prop Z-tests to compare to ...
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18 views

Covariance Problem

I'm trying to find the solution to $cov(y_i, \frac{\sum{y_i}}{n})$, where $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ (a regression model). Here's what I have so far... $cov(y_i, \frac{\sum{y_i}}{n}) ...
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20 views

For a random variable $X$, $M_x(t) = (1/81)(e^t+2)^4$. Find $\mathbb{P}(X \leq 2)$.

For a random variable $X$, $M_x(t) = (1/81)(e^t+2)^4$. Find $\mathbb{P}(X \leq 2)$. I know I need to somehow expand the mgf so then I can just sum the coefficients for $e^{0t}, e^{1t}, e^{2t}, ...
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How to generate Effect Size (SMD with inverse variance) in cluster trial

I've been working on generating an Effect Size from a Cluster Randomized Controlled Trial. The ES that I have generated are different from those reported in the paper. Could someone help me find my ...
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1answer
13 views

Equality of joint probability distributions

Suppose I have $X$ and $Y$ as random variables and have their joint probability distributions. Is the following correct: $f_{X,Y}(x,y) = f_{Y,X}(y,x)$ ?
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1answer
37 views

Do I use the Standard Deviation of my sample or the population to find the standard error.

A professor is interested in determining if attending college influences the level at which an individual cooperates with the police. The professor is unsure if attending college will teach respect ...
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2answers
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Understanding sampling from a normal distribution with zero mean

I'm studying probability. I came a cross "sampling from distributions". Given a probability density function $f_X(x)$, what I understood is that sampling means getting values of $x$ according to the ...
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1answer
30 views

How many times to roll a die before getting $n$ consecutive sixes given $m$ occurences?

Given a unbiased dice how to find the average number of rolls required to get $n$ consecutive sixes given already $m$ number of sixes occurred where $m\leq n$. I know how to solve using n consecutive ...
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0answers
19 views

Average of part of poisson distribution

Is there a way to find the average of part of poisson distribution? I tried it by finding the center of gravity of that part. However I faced the integration of factorial in the denominator of the pmf ...
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1answer
28 views

Standard deviation with exponential distribution

Let x denote the distance that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that x has an exponential distribution with parameter lambda = 0.01386. a. ...
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1answer
40 views

Find the probability generating function

I have an exercise of this type that I just can not solve "Are $x$ and $y$ be independent random variables, $X$-Poisson($a$), $Y$-Poisson($b$). Find the probability generating function of the random ...
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1answer
14 views

T statistic has t(n-1) distribution

I am trying to prove that $T_n=\frac{\bar{X}_n - \mu}{S/\sqrt{n}}\sim t_{n-1}$. One of the assumptions that seems to come up in proofs I saw of this is that ...
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2answers
58 views

Question on finding the MLE [closed]

I would appreciate any input or direction on this: I need to find the MLE of the hazard rate $\lambda$ where $$F(y;\lambda) = \begin{cases}1-e^{-\frac{y}{\lambda}} & y \ge 0, \\0 & y < ...
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1answer
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Get overall sum having partial data

I have partial data of online game ingame sales. I know the data for N users. The overall game users count is M which is about 6 times greater than N. I can calculate overall revenue for N users. But ...
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Measure Theory vs. Decision Theory - problem classification

I am having trouble classifying my problem, and I am seeking some guidance on book advice. I don't know if I have measure-theory problem and/or a decision-theory problem (or other field). I want to ...
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2answers
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Binomial probability on ports

This problem appears very simple, but I am almost positive that it should not be so simple. 10 ports. P1,P2,P3...P10 are connected to a computing device which polls them in order to check which ...