Questions tagged [statistics]

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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12 views

Finding Variance from given table using formula

Im having trouble calculating the variance for Xm and Xk for the following table: "We now consider a random household and let the stochastic variables Xk og Xm be defined in the following way: Xk:...
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7 views

Normal Distribution - percentiles

i have this closed question, I don't know how to solve it, I have only the answer but I would like to understand the solution and know how to solve them, Thank you! 1.From a population that is ...
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How to interpret minimal sufficient statistics?

I know the definition of minimal sufficient statistics is: A sufficient statistic T(X) is called a minimal sufficient statistic if, for any other sufficient statistic T'(X), T(X) is a function of T'(X)...
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1answer
20 views

Why expected value from a tries example $n=3$ is equal to expected value $n=654$. Why the population mean is still the same?

From a sample we have the expected value: $E[x] = μ$. And if we have differents trials $n=654$ the mean still is $E[x] = μ$. What is the reason of this relation? What causes this relationship to be ...
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19 views

How to remove skew from data?

I am trying to remove the effects from skew from data to find the true mean Let's say I a priori know that the data is drawn from a true Gaussian distribution, but I am unable to take an infinite (or ...
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1answer
21 views

Expected Value of MSR in General Linear Model

The setting of my problem is as follows: Consider the general linear model $Y = X\boldsymbol{\beta} + \epsilon$, where $\beta = \begin{bmatrix} \beta_0 \\ \boldsymbol{\beta_1} \end{bmatrix}$ and $X = \...
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2answers
31 views

How to compute probability of a bootstrap sample

The Question Consider the samples $\{1, 3, 4, 6\}$ from some distribution. a) For one random bootstrap sample, find the probability that the mean is $1$. b) For one random bootstrap sample, find the ...
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1answer
38 views

Find the joint pdf of $X_{(n/2)}$ and $M$.

Let $X_1, X_2, \ldots , X_n$ be a random sample from a Uniform$(0, θ)$ population with $θ > 0$, where $n$ is an even integer. Let $M=\frac{X_{\left(\frac{n}{2}\right)}+X_{\left(\frac{n}{2} + 1\...
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1answer
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Projected normal distribution: projecting 3D vectors perturbed by Gaussian noise.

I am interested in the following problem. Consider a unit vector $\hat{u}$ in 3D and the perturbed vector $\vec{v} = \hat{u} + \vec{x}$, where $x_i \sim \mathcal{N}(\mu=0,\sigma)$, that is, the vector ...
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1answer
27 views

How can I evaluate average with negative values, given I will apply this weight for both negative and positive values

I am more a programming guy than math one, so please be easy on me with math expressions Before I start describing to make it clear: I have 2 types of scores: reddit score and sentiment score. Reddit ...
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Book recommendation for formal, proof-based, theoretical books on time series and panel data

I'd like to ask you if there are any formal, proof-based and mainly theoretical books on time series and panel data. When searching for books on time series and panel data I got really good ...
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Sigma algebra generated by a set of functions

Let, $\Omega = \mathbb{R}^{[0,1]}$, that is the set of all functions from $[0,1]$ to $\mathbb{R}$, equipped with the sigma field $\mathcal{A}$ which makes each one dimensional projection measurable. ...
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How would you compute/test the conditions of the Gauss Markov Theorem?

I was looking at the Gauss-Markov theorem and its respective conditions but was confused by the notation. From my understanding, the Gauss-Markov theorem deals with the error term denoted as $\epsilon$...
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Sufficient Statistic problem

Let $\underline{X} = (X_1,X_2)'$ be a random sample from $\textrm{Po}(\lambda), \lambda>0$. Show using the definition that $S(\underline{X}) = X_1 + X_2$ is sufficient for $\lambda$.
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Finding conditional distribution in matching ordering situation

Suppose we draw two values $x_1,x_2$ according to a CDF $F$. Independently, we draw another two values $y_1,y_2$ according to another CDF $G$. Both $F$ and $G$ has support $[0,1]$. Among those four ...
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26 views

What does difference over sum mean in a set of values?

What would be the meaning of $X$ given that $A$ and $B$ are a list of values in which the values are approximately the same, i.e., $ \frac{A}{B} \approx 1 $ and both ${A,B} > 0$ ? $$ X = \frac{A-B}{...
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1answer
37 views

For valid probability densities $p(x)$ and $q(x), \int p(x)\ln(q(x))~dx = 0$. Can you catch my mistake?

I was trying to solve the following integral by parts, $$ \int p(x)\ln(q(x))dx$$ for valid probability density functions $p(x)$ and $q(x)$, $$ \int p(x)\ln(q(x))dx = \ln(q(x)) - \int\frac{q'(x)}{q(x)}...
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What is the probability that I will not get wet, given that it rains? [closed]

There is a .4 probability that it will rain today. There is a .32 probability that it will rain today and I will get wet. What is the probability that I will not get wet, given that it rains?
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4answers
52 views

The Variance Rule of $V(cX) = c^2V(X)$

As a statistics major, I learned a while back that the $V(cX) = c^2 V(X)$, where $V(X)$ is the variance of $X$. However, I never quite understood why that was the case. Consider this: We have an event ...
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1answer
30 views

Standard deviation (ni) impact

I would like to understand something when I use standard deviation to mesure my dispersion. Imagine at first series of prices $\xi$ : $1,2,3,4,5$ , the average price as $3$ and the sd as $1.58$. Now ...
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23 views

Undergraduate Statistics books suggestions

What are some topic wise good books to become a competent statistics graduate? The topics I've for my under graduation are 1)Descriptive probability 2)Probability distributions 3)Statistical methods 4)...
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2answers
25 views

Hypothesis testing - Find critical regions, at the 5% level of significance [closed]

The national proportion of people experiencing complications after having a particular operation in hospitals is $20\%$. A hospital decides to take a sample of size $20$ from their records. Find ...
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1answer
47 views

Is $\max(X_1,X_2,X_3)$ an unbiased estimator [closed]

Suppose that $X_1, X_2, X_3$ is a sample drawn from some population, if we use $\max(X_1, X_2,X_3)$ as an estimator of the population's mean, then can we say that its unbiased: My approach was to let $...
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17 views

Confidence interval for standard deviation implemented but unsure if correct.

tl;dr; why is does my monte carlo simulations differ so much with the confidence interval? So I was checking the confidence interval for the sample standard deviation for a normal distribution: https:/...
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1answer
20 views

$95\%$ confidence interval with unknown standard deviation

The Question We have the results for the heart beats per minute of a sample of $15$ people: $$54, 63, 58, 72, 49, 92, 70, 73, 69, 104, 48, 66, 80, 64, 77$$ Give a $95\%$ confidence interval for the ...
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23 views

Three different errors. Which one makes sense?

I have a question on clustering and I think MSE is the right place for it. My goal is to compare clustering methods considering different method and different number of clusters using an external ...
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2answers
40 views

Null and alternative hypotheses : Car model fuel usage [closed]

Some people assume that a specific car model does at least $\mu_0=120$ km with $1$ Lt of gasoline (petrol). $10$ independent tests give the following results: $$104, \ 96, \ 80, \ 100, \ 108, \ 100, \ ...
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3answers
29 views

How to find the degrees of freedom of the confidence interval for difference between means

You have a sample of 9 men and a sample of 8 women. Are the degrees of freedom for the t value in your confidence interval on the difference between means 16? How is this obtained?
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34 views

What sigma algebra are we using when considering the sum of two random variables?

If we consider random variables $X,Y$ both with the sigma algebra $F$, is $Z=X+Y$ defined on the sigma algebra $Z \times Z$ or on $Z$? In other words, if $X(x)$ is a function of x and $Y(y)$ is a ...
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9 views

EM Algorithm Shared Parameter Model

I am trying to find the joint model of a binary random variable $Y_2$ and a normal $Y_1$. I am planning to use a shared parameter model by introducing $b\sim N(0,\sigma_b^2)$ to induce correlation. ...
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22 views

Turning a PDF into a CDF in R [closed]

I was curious if it was possible to turn this probability density function into a cumulative density function? Also is there a way to print out the mean, standard deviation, quarterlies, etc. I have ...
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2answers
40 views

Help with S&P probability

According to this model, what is the probability that the change in the S&P $500$ on a particular day is positive. We have $P(X>0)$ and then converting to standard normal table I get $P(Z>(0-...
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1answer
24 views

Critical region for one-sided test of binomial proportion at 5% level

For this question's part C), the answer key says you must test the upper tail, but it isn't the alternative $p < 0.3$? So shouldn't you be testing the lower tail and get $x\le2$ for the critical ...
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38 views

books similar to '100 page machine learning book' but on probability and statistics

I would like to find a technical book that goes through the theory behind probability and statistics, although does not read like a textbook. Similar to the writing style of Burkov in '100 page ...
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15 views

Having a problem getting the Intuition behind t-tests and z-tests

Say we have a population of $n$ rodents and that the mean of the population is $\mu$ with standard deviation $\sigma$, the standard error of means in this population would be $\frac{\sigma}{\sqrt n}$ ....
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33 views

What does the summation in the 2nd term mean? Is the notation correct? (see the picture)

I specifically do not understand the i<j below the summation. Here is the application of the formula i am struggling with $$\sum_{i=1}^{j}p_i^{4}+\sum_{i=1}^{j-1}\sum_{i<j}^{j-1}(2p_ip_j)^{2}$$ ...
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1answer
23 views

What is the expected value of the estimator?

I would like to clear some doubts about estimators in statistics. By definition, an estimator $\hat{\theta}$ is a function that 'estimates' the value of the parameter $\theta$, itself being a random ...
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32 views

Multivariate Analysis Variability Question

For a random normal vector $y$, let $z=E'y$, where $E$ is a square matrix with eigenvectors of $\Sigma$ stored in the columns. I'm asked to compute the covariance matrix of the random vector $z$, and ...
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13 views

Probability of PCA parameters?

Let's say I have a PCA model computed using SVG and have kept $L$ parameters, so I have matrix of mean values $\mu$, a matrix with the components $W$ and a vector of standard deviations $s$(?). I can ...
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16 views

Suppose that we have n independent observations (X_i;Y_i) pf the random vector (X;Y) with pdf f(x;y) [closed]

How to prove the following expression and what does the notation before variance mean?
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1answer
19 views

Sufficiency and completeness in two distribution

I have two variables $X_1$ and $X_2$ which follows exponential and double exponential with rate $\theta$ $X_1\sim\frac1\theta \exp((-x/\theta)),\quad X>0$ $X_2\sim\frac2\theta \exp(-2x/\theta)),\...
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8 views

Deriving estimator for non gaussian linear model

Gaussian linear models are often insufficient in practical applications, where noise can be heavy- tailed. In this problem, we consider a linear model of the form yi = a · xi + b + ei. The (ei) are ...
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How to get an estimated number depending of actions

I have a question and is if is there any formula or any way to calculate this : Example : I have an account, where I have number of posts, and number of likes. So is there any way to get an ...
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15 views

Independent events with change of origin

if we change the origin of 2 linearly independent variable will they continue to be linearly independent? I have a problem where i need to change the origin I wonder if changing origin will change my ...
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3answers
42 views

Uniform Distributions Ratio [duplicate]

Let $\xi$ is $U(0, 1)$, $\nu$ is $U(0, 1)$. What type of distribution is $g = \frac{\xi}{\nu}$? I have build logarithm of $g$, here is the plot but it seems like it is not normal distribution: ...
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2answers
33 views

How to solve this Confidence Level Question?

A company has conducted a test on $400$ people. Out of $400$ people, $224$ people like chocolate. Find the percentage of people who like chocolate in the $99\%$ confidence interval. The problem is ...
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1answer
24 views

Prove whether the M.o.M. estimator is consistent or not

Let $X_{1}, X_{2}, \cdots, X_{k}$ to be random samples from $Binomial(n, p)$. And denote k'th moment of distribution as $\mu_{k}$. Then the first and second moments are: \begin{align} \mu_{1} = np, \...
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24 views

I'm writing a probability and statistics paper on poker. Any ideas on probability theorems/ideas that i could incorporate into the paper? [closed]

I'm writing a paper attempting to apply probabilities and statistics to poker interactions. I've used combinations and conditional probability(Bayes theorem) but my paper still lacks substance. Are ...
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1answer
20 views

How can you calculate the probability of failure of a product after a given time, given the expected failures per unit time? [closed]

For example, if a product has an expected failure rate of 1 failure every 10 years of operation, what's the probability that the product fails after 1 year of use? How does this probability change if ...
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15 views

Transform a Two-Tail Requirement into a One-Tail Requirement

The Setup Let $X$ be a random variable with an unknown distribution. Let $p$ be the probability that $X$ is between $a$ and $b$, where $a<b$. Transform this Two-Tailed requirement on $X$ into a ...

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