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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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How can I derive the waiting time distribution of gaussian processes?

Given two successive Gaussian events, $X_1 \sim N(\mu,\sigma^2),\qquad X_2\sim N(\mu,\sigma^2)\qquad$ with $X_2 = X_1+\epsilon$,$\quad\epsilon\in {\rm I\!R},\quad\epsilon>0$ how can I derive the ...
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28 views

How to find Mu when only given two sample means

I need some help with the following question: I cant figure out how to solve this one because I only have sample mean but not population mean. Questions I have encountered like this previously had ...
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1answer
29 views

Calculate $E(Y)$ where $Y=X^{1.5}$

Let $X$ be a rv which is $Exp(\lambda=2)$. The pdf of $X$ is given by $f_X(x)=2e^{-2x}, x\geq 0$ (and $0$ otherwise). We define $Y=X^{1.5}$ and ask $E(Y)$. It does not look like the moment ...
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Calculate the confidence interval of $u_x-u_y$

Chemically it is established that the percentage of iron in 5 samples of ore extracted in a certain mine has an average of 22.3% and a standard deviation of 1.8%. In 6 samples of extracted from ...
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Producing a range of 0-100 based on multiple variables with different units (percent, numerical)

I'm absolutely no statistics wizard and it is my first post here, so I hope you bear with me. I'm working in Excel on a project where I need to convert data from a number of variables into a "one ...
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26 views

show $X = \frac{d_2Y}{d_1(1-Y)}$ has $F$ distribution if $Y$ has $B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)$ distribution

I search internet alot but I didn't find any prove for this theory. if $Y$ has $B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)$ distribution, then show that $X$ with given formula has $F(d_1,d_2)$ ...
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Aggregation of Surveys

I am doing an analysis of employee performance. And have two questions regarding sample size. (All calculations come from Survey Monkey’s Sample Size Calculator) Say an employee performs 500 tasks ...
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2answers
82 views

Conditional expectation, what is my mistake

From SOA sample #238: In a large population of patients, $.20$ have early stage cancer, $.10$ have advanced stage cancer, and the other $.70$ do not have cancer. Six patients from this population ...
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5 views

Consequence of assumption violation for chi squared [on hold]

I am fairly new to statistics and I've run into a problem in an assignment. I have to explain the consequences of violating the assumption that X follows a normal distribution when testing on a single ...
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Understanding the method of obtaining an expression for the $p$-value using a test with test statistic $T =\sqrt{n}(\bar{X}_n - \theta_0)/\sigma$

We are in a setting where we $X_1,\ldots,X_n\sim N(\theta, \sigma^2)$ where we assume $\sigma^2$ to be known. We test $H_{0}:\theta\leq\theta_{0}$ versus $H_{1}:\theta>\theta_{0}$. I am aware that ...
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If T is a Consistent estimator of $\theta$ then $T^3$ is consistent estimator of $\theta^3$

If T is a Consistent estimator of $\theta$ then $T^3$ is a consistent estimator of $\theta^3$. Can anyone tell me in a single line what's the logic behind this? I am familiar with invariance ...
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Sample size and margin of error in a confidence interval for a mean

I am having trouble understanding this problem: Darren wants to estimate the mean age in a population of trees. He'll sample $n$ trees and build a $90\%$ confidence interval for the mean age. He ...
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Ameba splitting probability (how many of them will be if now there are n?) [on hold]

The problem is quite common, but I cannot come up with an answer to this situation. We have n amebas. Each minute each ameba can die with probability 25%, stay alive without changes - 25%, or split ...
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Connection of the critical value for KS distance and model complexity

I wonder about a correlation of the critical value for KS distance (with Lilliefor correction) and model complexity? The critical KS value is distribution-independant if we use the true model ...
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2answers
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Normal Distribution in Probability

I can solve the k, but I am not really able to do the rest! The mass M of apples in grams is normally distributed with mean μ . The following table shows probabilities for values of M . Values: ...
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To find the p values in hypothesis testing

The null hypothesis says that at least 20% of college students are left-handed. If we took a sample of 20 college students and let $X$ be the number of lefties in the sample. Calculate the p values if ...
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1answer
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Suppose that $f (x, y) = xe^{−x(y+1)}$, where $0 ≤ x < ∞$, $0 ≤ y < ∞$. Find marginal densities

This question comes from rice 3.14 Suppose that $$f (x, y) = xe^{−x(y+1)}$$ where $0 ≤ x < ∞$, $0 ≤ y < > ∞$ a. Find the marginal densities of X and Y . Are X and Y independent? b. ...
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1answer
26 views

Divergence measured from integral of minimums: $\int_{-\infty}^{\infty} \min(p(x), q(x)) dx$

I was studying a particular machine learning algorithm (the GeoGAN) and although this isn't mentioned in the paper, it seems to be that under certain conditions, the algorithm is maximizing (over $p$) ...
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1answer
12 views

finding marginal distribution - how to determine the limits of integration

This exercise comes from Rice 3.12: let $f_{XY}(x,y)=c(x^2-y^2)e^{-x}, 0\leq x <\infty, -x \leq y \leq x$ b) find the marginal densities I have found thatc $c=\frac18$ and also that $f_X(x)=\...
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1answer
43 views

Let $X_1,X_2, X_3$ be a random sample from Bernoulli distribution. Which of the following is sufficient for P.

Let $X_1,X_2, X_3$ be a random sample from Bernoulli distribution $B(p)$ .Which of the following is sufficient statistic for $p$ $(A)\ \ X_{1}^{2}+X_{2}^{2}+X_{3}^{2}$ $(B) \ \ X_1+2X_{2}+X_{3}$ $(...
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when is product of rectangular functions not equal to zero given different centers?

I'm self-studying math, and came across a problem: $$ \prod_{n=0}^{N-1} u[x_n + a] - u[x_n-a] $$ where u is a unit step function, $x_n$ are sample points, $a$ is a constant. So basically the centers ...
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Showing Rejection Region Equality with Fisher Distribution

I'll preface this with saying that this question is a homework question, but not one that is graded or turned in in any form. Specifically it is from the text Mathematical Statistics with Application ...
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$E(min(G,G'))$: expectation of minimum of iid Gamma(m,1) distribution without calculus

I'm quite stuck on this. If $m=1$, both are exponential. Then min is quite easy to get. I also know that $E(max(G,G')+min(G,G'))=E(G+G')$ which is again known. But I'm not sure what else I can do in ...
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38 views

Proof of a Probability Problem by Induction

Dear Statisticians and Mathematicians, I am interested in proving the following lemma by induction I have shown that it holds true for $n=2$ which I don't provide its proof here. We assume it is true ...
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What kind of statistical test should I use to compare a variable between two groups? (Problem i found at work)

I work for a bank in south america and we do something called elasticity price test to determine what is the best "Price" (fee for a product), here is how it goes: We select two similar groups and ...
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12 views

Statistics - Question on Sampling

Here's the question The scores, $X_1$ and $X_2$, in papers $1$ and $2$ of an examination are normally distributed with means $24.3$ and $31.2$ respectively and standard deviations $3.5$ and $3.1$ ...
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Binomial distribution calculate probability of pikes - is my solution correct?

In a lake, there are two types of fish: trouts and pikes. Let p = 0.7 be the proportion of trouts in the lake. We pick 20 fish at random with replacement. Let X be the number of trouts. a) What ...
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1answer
24 views

Different definition of the geometric distribution

From SOA Sample #199: A company has five employees on its health insurance plan. Each year, each employee independently has an $.80%$ probability of no hospital admissions. If an employee ...
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1answer
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Simulation: Generate random numbers that cluster around an average?

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
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Marginal normal distribution from joint distribution defined on the Malta cruise

Given $X$ and $Y$ random variables with joint distribution: $f(x,y)=2\phi (x) \phi (y)$ defined in: $0 \leq y \leq x < \infty$ $ - \infty < x \leq y \leq 0$ $0 \leq -x \leq y < \infty$ $...
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Mean of the product of order statistics.

Let $X,Y$ be two real valued random variables with first and second moments. Let $(X_1,\dots,X_n)$ i.i.d $X$ and $(Y_1,\dots,Y_n)$ i.i.d $Y$ be to independent $n$-samples. Denote $\left(X_{(1)},\dots,...
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1answer
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Show that squares of Pearson coefficients are equal

Show that in SLR, the square of Pearson coefficient between $y$ and $ŷ, r^2_{y,ŷ}$, is equal to the square of Pearson coefficient between $x$ and $y$, $r^2_{x,y}$ I computed the least square ...
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1answer
19 views

Probability question, please please help

A box contains 18 balls. 6 are black and 12 are white. A sample of 4 balls are taken with replacement, what is the probability that the sample contains no more than 2 black balls I would really ...
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2answers
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Probability that a loss occurs to a randomly chosen customer of an insurance company - is my solution correct?

An insurance company divides its customers into different risk groups. Suppose that 60% of the customers are in group A (low risk), 30% are in group B (medium risk) and 10% are in group C (high risk). ...
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1answer
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Find a test for $H_{0} : \sigma_{1}^{2} \ne \sigma_{2}^{2}$, against $H_{1} : \sigma_{1}^{2} =\sigma_{2}^{2}$

Consider $X_{1}\dots X_{n} $ ~ $N(a_{1},\sigma_{1}^{2})$ and $Y_{1}\dots Y_{m} $ ~ $N(a_{2},\sigma_{2}^{2})$, and they are independent. We need to find a criteria for $H_{0}:: \sigma_{1}^{2} \ne \...
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A Conditional Probability Problem

I am interested in finding the following problem: Let $\tau_1$ and $\tau_2$ are ordered statistics from a set of 2 independent uniform $(0,t)$ R.V. and let $Y_1,Y_2,Y_3$ are nonnegative iid R.V. that ...
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1answer
36 views

Probability that at least 1 machine breaks down

In the manufactoring plant of a company two large production machines are used. Machine A breaks down with probability of 0.01, whereas a breakdown of machine B occurs with probability 0.005. Further, ...
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1answer
27 views

Random variables with arbitrary positive correlation but without arbitrary negative correlation. Then $Cor(X_i,X_j)<\frac{-1}{n-1}$ not possible

Let $X_i \in L_2$ be a sequence of pairwise correlated random variables. The random variables can have arbitrary positive correlation but can't have arbitrary negative correlation. How can I show ...
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23 views

Random Sequential Adsorption Paper: Probability or Probability Density?

While going through a paper on Random Sequential Adsorption ("Dynamics of polydisperse irreversible adsorption: a pharmacological example" by Erban et. al, 2007, https://www.worldscientific.com/doi/...
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Probability Distribution table [on hold]

In a key ring there are 3 keys. X is the number of keys chosen before opening the door. Make the probability distribution table
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1answer
21 views

fint joint and marginal distributions of two uniformy distributed variables over a specified region

This exercise (partially) comes from Rice 3.9 Suppose that (X, Y ) is uniformly distributed over the region defined by $0 ≤ y ≤ 1 − x^2$ and $−1 ≤ x ≤ 1$. Find the marginal densities of X ...
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1answer
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Find the marginal distribution of an point randomly chosen on an ellipse

This exercise comes from rice 3.6 and states: A point is chosen randomly in the interior of an ellipse: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Find the marginal densities of the $x$ and $y$ ...
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Question about gamma and inverse gamma prior

Suppose we had an expressions that was proportional to $$\frac{1}{\sigma^{3}}\exp(\frac{-\beta}{\sigma^{2}})$$ My question is, can you choose different possible parametrization for a prior ...
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1answer
19 views

Moment generating function of a binary variable

We have a set of Random Variables $Y_i$ which takes the value $\alpha$ with probability $(1-p)$ and takes the value $1-\alpha$ with a probability of $p$. We have been tasked with finding the Moment ...
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35 views

A simpler proof for a probability problem

I am trying to prove the following lemma 2.3.5 from Stochastic Processes, Sheldon Ross, 2nd ed, page 77 which I have provided it here for convenience. The proof is provided in the book based on a ...
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1answer
20 views

Bernoulli trials hypothesis test

I'm studying about hypothesis test theory and I've reached the next exercise: Let $X\sim Ber(\theta)$. Then, we know that for a sample of $n=10$, we reject $H_0:\theta = 0.5$, and accept $H_1: \theta ...
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1answer
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We choose 2 different people - what is the probability that they all have the same color of the eyes

So I have big problems with the problem above... I tried for like 5 hours to find the solution but I just don't know how to proceed. So, there are 45 people, and 5 eye colors. Dark Brown = 20; Blue = ...
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24 views
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How to find the probability of a 'mean shift' in a random process?

How would I go about calculating the probability that the observed mean of a collection of data generated by a random process, which is governed by a particular statistical model, would deviate from ...
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calculate sample size for specified type II error probability, comparing 2 proportions

This is from Devore Probability and Statistics for Engineering and Sciences 9th edition, chapter 9, section 9.4, exercise #53 on p.398. The problem involves a hypothesis test comparing two ...
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Branching Processes Extinction Probabaility [on hold]

A cell produces 0, 1 or 2 offspring with probabilities 0.2, 0.2 and 0.6 respectively. (You may assume that cells produce offspring independently of one another.) What is the extinction probability ...