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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3-sigma approximation

I am making a system involving a sensor who has to be really precise. I found on their datasheet a diagram that shows the typical performance of the sensor. There's the mean value, the +3 sigma, ...
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38 views

Doubt about a probability excercise

I'm a statistics teacher at a college. One day a student came with a doubt about an excercise about probability. The text goes like this: A person has two boxes A and B. In the first one has 4 white ...
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18 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
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what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula, and learned shortly thereafter that it was derived from this formula. ...
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48 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
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11 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
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2answers
27 views

defective component and probability

An electronics industry uses three plants from A, B and C in the ratio of $3$ to $2$ to $1$. However $1\%$ of components manufactured by A, $2\%$ of components manufactured by B and $3\%$ of ...
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26 views

Games and statistics

Three individuals A, B and C alternate in contention of a game according to the following rules: A plays with B and the winner plays with C. The game continues until one of the individuals to win two ...
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5 views

Concentration bounds on Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...
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1answer
9 views

Binomial Distribution formula

If $X\sim B(n,p)$, show that $P(X=r+1)=P(X=r) \cdot \frac{p(n-r)}{q(r+1)}$ for $r=0,1,...,n-1$ My attempt, $P(X=r+1)={_n}C_{r+1}(p)^{r+1}(1-p)^{n-(r+1)}$ How to proceed then?
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16 views

Calculate pdf and cdf of exponential random variable

I am not looking for an answer as this is for a homework exercise i just want to be pointed in the right direction so i can learn how to do it. The question is: Let X ∼ Exp(λ). Calculate the cdf and ...
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22 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
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1answer
12 views

Summatory problem | Ordinary least square estimator

How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}} \\ & = ...
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1answer
42 views

Rumor and probability

31 people in a community, a person has a rumor to a second which, in turn, repeats to the third, etc. At each step the person receiving the rumor is randomly chosen among 30 people available. a) Find ...
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2answers
39 views

Two urns, A and B, each with two drawers.

If you have two urns, A and B, each with two drawers. The urn A has a gold coin in a drawer and a silver coin in the other drawer while the urn B has a gold coin in each drawer. An urn is chosen at ...
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20 views

Wick's theorem: Classical Version. Derivation question.

I am trying to prove the classical version of Wick's theorem: For a set of random variables ${a_i}$, with covariance matrix $M$ and $\rho(\vec a)$ a Gaussian probabilitiy density: $$\langle a_j a_k ...
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4 views

assumptions of t-test of correlated terms [on hold]

Good Day ,i just want to ask if what are the assumptions of t-test of correlated means.I've been browsing the internet since yesterday because im just so curious if what does this statisticall tool ...
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28 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
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1answer
18 views

Probabilty of Drawing Specific Cards

If I have a deck of randomized cards and I draw 5 cards, what is the probability that I will draw at least one 2 and at least one 3. In other words, I am looking for any hands of the form x2xx3, ...
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28 views

What is the intuition behind slovin's formula?

Here is the formula. $n=\dfrac{N}{1+Ne^2}$ I don't understand why this equation works. What value does $1+Ne^2$ represent?
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5 views

Fitting power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
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1answer
35 views

“Well known properties” of Poisson distribution

I'm working with Bradley Efron (2010): Large Scale Inference and my question concerns the proof of Lemma 2.3. Here we have $z_i \sim F_0$ with probability $\pi_0$, $z_i \sim F_1$ with probability ...
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34 views

Finding Probability of picking one ball out of N balls.

presented with n identical balls, one with a prize in it. Picks each ball out idependently one at a time till gets prize. I need to find the mean and variance of the number of balls needed to pick ...
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6 views

Success ratio improvement

I got the following question from a friend: Suppose the success ratio improved from 98% to 99.5%. What is the ratio of the improvement? This is the answer I gave him which he deemed correct: 99.5% ...
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42 views

Product of +1 and -1 with all combinations

I am looking for an algorithm or a smart way to do this in excel. I have this table. ...
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18 views

cluster sampling methods and stratifying sampling method

Briefly discuss one similarity and one difference between cluster sampling methods and stratifying sampling method.
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28 views

Expected value of trials to obtain a red ball in a box of white balls.

I have a problem that involves a box containg N balls, one of which is red and the rest of which (N-1) are white. The question involves finding the expected value and variance for the number of trials ...
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37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...
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2answers
47 views

Solve $\frac{1}{2^\theta}\sum_{k=0}^{\theta} {\theta\choose k} \delta(k)=\theta$ for $\delta$

The following arises in unbiased estimation of a parameter for the binomial distribution, but that information is not needed for solving the question. I tried solving this by manipulating the sum to ...
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1answer
23 views

Biased sample from biased sample

A webpage has users, where each user has a number of projects uniquely assigned to him or her. I want a random sample of users by randomly sampling projects and then taking the users connected to this ...
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1answer
26 views

How can I calculate a campaign goal ongoing?

I work for a non-profit and we are continuing a campaign to serve 1000 children at $29 a month. As of today we have the funds to serve 398 children based on an average from the last 12 months. My ...
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6 views

Testing Relative Preference Fit (multi-variable)

I'm playing around with a data set that I have built which is meant to represent the subjective preferences of a person related to a particular product. I want to compare, then, the properties of a ...
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2answers
39 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
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8 views

Repeated Measures…(Analysis of Variance) [on hold]

What are within-subject variables and what are the difference scores between these variables referred to?
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Estimating Johnson Distribution Parameters by Quantile

In this paper: http://www.researchgate.net/publication/31291960_Quantile_Estimators_of_Johnson_Curve_Parameters the four parameters for the Johnson distribution ...
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11 views

the meaning of 4-wise hash function

If someone says: 4-wise independent sign (hash) functions $s_1,s_2, s_3 : [d] → \{+1, −1\}$, then what does it means? I cannot use k-wise Independence variables (the definition 1 ...
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Hypothesis Testing with given Pearson correlation.

There are two independent sets of samples , female and male. The problem is to calculate the 95 % C.I. of the mean of total( male+ female, two samples) population. -> Female : n=100, mean= 169.1, sd ...
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1answer
17 views

How to find the degrees of freedom for a chi-square variable

How does one find the degrees of freedom for a Chi-square random variable when trying to fit a distribution to a sample? I read an explanation regarding this in this source. I don't understand how to ...
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23 views

I want to know my hypothesis is right

Could you determine does the following formulation makes sense? I wanted to prove that the "Mahalanobis distance ($\mathbf{(x_i - \bar{x})^T \Sigma^{-1}(x_i - \bar{x})}$)" is a Log Likelihood Ratio ...
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1answer
11 views

Do the bounds change when multiplying a probability distribution by a constant?

Do the bounds change when multiplying a probability distribution by a constant? For example consider when an exponential distribution bounded by upper bound A, lower bound B and mean X, is multiplied ...
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48 views

Need simplified formula of probability equation

I have RV $x$ which is function of independent continuous RVs $x_1$ and $x_2$. After some manipulations, I came up with an expression for the outage probability of $x$ as $$P_\text{out}(y)=\Pr(x\leq ...
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34 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
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1answer
22 views

Univariate Normal Distribution Transformation

X is the time taken for a high school athlete to complete a 100m sprint, in seconds, which follows a normal distribution with a mean of 16. The athlete would complete the sprint under 19.495 seconds ...
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Cost per pound savings analysis between two data sets

Given 2 sets consisting of (type, cost, weight) and grouping the data into sub-sets based on type (for simplicity let's just say their are two types). How would you go about comparing these data sets ...
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32 views

black jack variance

Is it correct to compute variance in Blackjack using Bernoulli distribution? For example: $p = 41\%$ is the probability of winning $q = 9\%$ is the probability of tieing $t = 50\%$ is the ...
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1answer
23 views

Example of a distribution with mode undefined

Is there any distribution with an undefined mode? If all the elements of the data set have frequency 1 is the mode undefined or all the elements are modes of the distribution?
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probabillity and statisstics [on hold]

Thirty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 70% can be repaired, whereas the other 30% must be replaced with new units. If a company ...
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+300

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
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2answers
44 views

Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
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How to solve this confusing iterative limit equation

I have this equation which is intended to be used on a data set of integers. I am attempting to create a software algorithm using this equation, but I am very confused about how to implement it. ...