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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Statistics - Estimation problem

I am struggling with a statistics problem that seems quite easy but don't know what to do. In a factory a product is given to two experts - X and Y. They have to independently test the product and ...
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1answer
27 views

Joint PDF given, find missing constant

I have a relatively simple question regarding a joint PDF given to me. There are 2 random variables X and Y, with the following joint PDF. a) Find the value of a. I have attempted to set up the ...
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0answers
13 views

Achieving the Cramer Rao Lower Bound (CRLB)

can you kindly explain why is this so? I have obtained as the MLE to a parameter $\alpha$ as $\hat{\alpha} = \frac{\bar{Y}}{k}$. This is an unbiased estimator for $\alpha$. Supposing the score ...
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0answers
24 views

How to calculate the z-score for the number of times an event occurs?

I have a set of known data, assumed to be correct, and a set of new data that I want to check for falsification before merging it into known data. The data is essentially the probability of an event ...
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1answer
20 views

Calculating the number of arrangements of items into compartments?

I'm dealing with intro probability theory here, and am a bit perplexed by the logic around arranging items in boxes. When arranging a small set of objects in a series of boxes, I am not seeing why the ...
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1answer
18 views

Finding the variance of X with a variable p between 0 and 1

Here is the full question (and note this is a question on my homework, but I just don't understand a portion of it): Suppose a random variable $X$ has the support of $S_X = \left\{0,1\right\}$ with ...
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0answers
12 views

Regarding the distribution of pivotal functions not depending on their parameter(s)

I have difficulties understanding the part of pivotal functions not depending on their underlying parameters. Let's take a simple example, if Y is a random sample from an $N(\mu,1)$ distribution and ...
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0answers
16 views

Intuition for “weights” in simple linear regression [migrated]

Suppose we have data $\{x_i,y_i\}_{i=1}^n$ where $x_i \in \mathbb{R}$ and $y_i \in \mathbb{R}$ and we model $$ y_i=\beta x_i + \varepsilon_i $$ The ordinary least squares estimate of $\beta$ is $$ ...
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0answers
8 views

What do I enter in 'number of successes:' for Confidence Interval Proportion (excel)

I am trying to calculate the confidence interval proportion in Excel and it comes up with a pop up box asking for the sample size, which I entered, and the 'number of successes', which I dont ...
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2answers
24 views

Find the population mean and standard deviation

Divide the population of U.S. workers into two categories: those who have a job in the goods-producing industries, and those who have a job in the service-producing industries. Among all workers, ...
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0answers
35 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
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2answers
24 views

Use of permutations and combinations for a conditional seating arrangement

I've got a problem where there are 6 seats in a row, with 6 people to be seated. I need to figure out all possible seating arrangements, given that two people are married and are to be seated next to ...
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0answers
20 views

Calculate trend of week

Lets say I have the statistics for the crimes that were committed this week in California: On Monday there were 10 crimes committed. Tuesday: 6 Wednesday: 7 Thursday: 15 And on Friday: 8 Now I ...
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0answers
38 views

Probability distribution of request handling

I have values representing time taken to execute one request on server. Could somebody advise what type of distribution it is? I think that normal distribution but I am not really sure about it. ...
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1answer
37 views

Dependent Bernoulli trials confidence interval

I would like to know if there is a way to build a confidence interval, for a random variable which has a Bernoulli distribution, based on its history. I mean if the order of its states is 11100 (i.e. ...
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0answers
23 views

Importance sampling estimate

Suppose that one is interested in estimating the tail probability $$ P(Z \ge b) $$ for $$Z \sim N(0,1)$$ and a large threshold b. What is the expression for the importance sampling estimate with ...
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0answers
18 views

two different Monte Carlo approaches

Assume that the function $f$ is integrable and maps $[0, 1]$ into $[0, 1]$. Consider estimating $\int_0^1 f(x)\,dx$ using two different Monte Carlo approaches. The standard approximation is applied in ...
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0answers
19 views

Statistics: Sampling without replacement

A subdivision of a town has six homes, one with 1 bedroom, one with 2 bedrooms, one with 3 bedrooms, one with 4 bedrooms, one with 5 bedrooms, and one with 6 bedrooms. A potential buyer chooses at ...
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0answers
51 views

Fastetst method for calculating $\frac{(a+b)!}{a!b!}\bmod{m}$

Is there any faster method for calculating $\frac{(a+b)!}{a!b!}\bmod{m}$? Lucas theorem is also turning out to be slow! $a,b\leq10^9$ and $m=10^6+3$.
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1answer
44 views

Approximation for the integral $\int_0^{\infty } \frac{x}{(a x+1)^2} \, dx$

I want to find the mean of random variable $x$. In my analysis, I came up with an integral $$I=\int_0^{\infty } \frac{x}{(a x+1)^2} \, dx$$ which does not have a closed-form solution, probably it ...
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4answers
39 views

A fair die is rolled three times, find the probability of the following events:

a. All rolls show an even number of dots b. the last two rolls show an even number of dots c. the third roll shows an even number of dots d. every roll shows a single dot e. every roll shows the ...
3
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1answer
22 views

Moment generating function of Random Sums

I am unsure of a particular step in the supplied solution of this problem. Problem: We are given $X_{i}$, for i = 1,..., n, is a sequence of iid Geometric Random Variables. N ~ Geometric(p), and ...
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0answers
27 views

Two-parameters Wiener process

Two-parameters Wiener process $W(r, u), r \in [0, 1], u \in [0,1]$ is a stochastic process with a covariate kernel $\mathbb{E}\left[W(r_1, u_1) W(r_2, u_2)\right] = \min(r_1, r_2) \min(u_1, u_2)$. ...
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1answer
24 views

Finding the CDF of a piecewise PDF

So for my statistics class I am taking this semester we've been working on continuous random variables and we have one question that the teacher did not cover at all nor his notes, and it has to deal ...
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1answer
11 views

find the Expected Value: (1/n^2)E[(ln(x1)+…+ln(xn))^2]

I need to find the Expected Value: (1/n2 )E[(ln(x1)+...+ln(xn))2 ] and the only information I get is that E[ln(xi)] = a, E[ln(xi)2 ] = 2a2 where i = 1,2,...,n. The problem I'm having is with the ...
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0answers
11 views

GLM for normal distribution

$Y_i$~N$\left(\mu_{i,}\sigma^2\right)\space \mu^2=\alpha+\log \left(\beta_0+\beta_1x_i\right)\space \alpha\space is\space unkown$ how is this proved to be a Genralized Linear Model? My assumption ...
0
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0answers
21 views

Inverse Square Root Of Matrix

So let's say a matrix is A. Then how do you find A^-1/2? It seems to be different from finding the inverse of A. Could someone provide a simple example as ...
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0answers
51 views

Exponential moments and integrability

Given a probability distribution $P(x)$ for which the moments $\int x^rdP(x)$ and the exponential moment $\int e^{Rx}dP(x)$ exist for some positive integers $r$ and $R$, than is it true that ...
1
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1answer
15 views

Paired and unpaired data-Statistics/Hypothesis testing

I'm getting a bit confused about paired and unpaired data, for example in this question I don't understand how this data is paired. If it was the same steel pipes that were left uncoated first in ...
0
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0answers
12 views

Statistical tolerancing of 2 independent, normally distributed, variables. What is the reliability probability?

The image below is of a solid, machined part. Ignore everything aside from the red, that is where a slot has been cut to fit a block. With the assigned tolerances what is the probability that a ...
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0answers
22 views

Statistics - 95% confidence level?

You are to conduct a sample survey to determine the mean family income in a rural area. The question is how many families should be sampled? In a pilot sample of $10$ families, the standard ...
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1answer
31 views

Normal Distribution: Covariance of sample mean and sample

$X_1,\ldots,\overline{X}_N$ is a random sample of $N(\mu,\sigma^2)$ observations. Let $\overline{X}_N$ denote the sample mean. What is $\operatorname{cov}(X_1, \overline{X}_N)$? Since the mean of ...
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0answers
8 views

Stability of boostrap confidence intervals

As a word of background, I want to show that certain result is stable when averaging over a large number of simulations, but could be just a lucky draw with a small number of simulations. I have a ...
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0answers
16 views

Systems of equations that are over and under-determined at the same time

I know what an over and under-determined system is. What I am not sure is how to show both. I am asked to come up with a matrix that is both an under-determined and over-determined system which is the ...
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0answers
17 views

Conditional expected waiting time for a bus to arrive given number of people waiting

Suppose there are $N$ people and each person arrives at the bus stop after a waiting time distributed i.i.d. as an exponential distribution with parameter $\lambda$. Suppose further that the bus ...
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1answer
25 views

normal approximation of binomial distribution

a school buys 60% of its light bulbs from supplier A and 40% from supplier B. the light bulbs from both suppliers look identical but light bulbs from supplier A have exponentially distributed ...
3
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1answer
34 views

How can I calculate Index of Coincidence of Vigenère cipher?

I have computed the letter frequency of the cipher text. However, I don't know how to apply Friedman Test to Vigenère cipher. I couldn't calculate the Index of Coincidence. Does anyone can help to me ...
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2answers
40 views

Over-determined and Under-determined systems

How do I show that a system is both over-determined and under-determined? I am supposed to come up with a matrix that satisfies both but am not really sure I understand what types of equations would ...
4
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1answer
71 views

Integrate two sets of Data and check similarity

I have the following CSV Data I used Excel Charts to plot the Data I want to compare these Data with other Data, in another word i want to comapre 2 curves and check the similarity between them in ...
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1answer
28 views

How can we terminate in a finite amount of time?

What's the difference between: 1.Always terminates in a finite amount of time 2.Terminates in a finite amount of time with probability 1 Extension: Are there any more ways we could add to the list? ...
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0answers
17 views

Scratch card odds

I'm trying to work out how to set up a scratch card system with a preset probability. Lets say I have types going from 1-8 which corresponds to your chance of winning. So a type 8 card is the ...
0
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2answers
28 views

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y?

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?
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1answer
22 views

Statistics - Mean value and net revenue

From the following book:Probability and Statistics for Engineering and Science
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1answer
35 views

Identifying Break-Points in Binomial Responses (Alternative to Multiple Chi-Squared Tests?)

So I have a research question whereby I have subjects' age and a yes or no response to a question. Ages range from 45-90 approx. I am currently looking at whether age has an affect on these yes or no ...
0
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0answers
19 views

how to find $X^2_{0.95,14}$, $P(X^2_{9} \geq y) = 0.99$ in a TI-84 calculator?

Can someone please show the steps in how to find $X^2_{0.95,14}$ in a TI-84 calculator? My professor does not want us to use a statistics table, but to know how to use the calculator in quizzes and ...
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2answers
30 views

statistics dice problem

If $5$ fair dice are thrown at the same time, how do you find the probability that there are three $1$'s and two $2$'s? The answer says its $5C2 \cdot (1/6)^5$ but I don't understand why.
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1answer
37 views

chose the best path for estimation

I have a Cartesian grid (100x100) in which some of the points are known (30 out of 10,000) and the rest are unknown. I want to use the known points and estimate the other cells. Is there any ...
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0answers
27 views

Gaussian distribution finite population with unknown cardinality

I have taken a sample population of a population with unknown size. The sample size is 54 trades. The sample mean is 2.1% (1.021) return per trade. The sample standard deviation is 0.01. 100% of ...
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1answer
25 views

monte carlo simulation: multiple variables

I am reading about the merton model in finance. It depends on multiple distributions. I want to use monte carlo simulation but I've a little question about this. If we have a statistical model that ...
0
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1answer
16 views

standard error for the parameters of a linear regression model

Given a linear model $\mathbf{y} = \beta \mathbf{X} + \epsilon$, it is well known that the estimate for $\beta$ that gives the minimum residual sum of squares (RSS) is given by $\hat{\beta} = ...