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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Confidence interval for a normal mean when the variance is unknown

For this problem , how to calculate when $t.025,8 = 2.306$?
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208 views

Statistics: Combination problem

Good day everyone. I'm doing a combination problem and its solution keeps eluding me. Problem: Of a hand of 13 in a deck of 52 how many combinations are there of none of the cards being greater ...
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4k views

Is it possible to calculate the mean and standard deviation from a median and quartiles?

Any advice would helpful. I understand that the reporting of median and quartiles for small samples is an indication of skewed data. If such is correct, then is it useless to try to work out the mean ...
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96 views

bernoulli trial

I have to find a probability that 2 in 30 electrical component coming off a production line is defective. The probability to get a defective one is 0.04. Assume that whether or not a component is ...
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144 views

Expected value for $f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$

$$f(x)= \frac{\Gamma (\alpha+\frac{1}{2})}{\Gamma (\alpha)} \frac{\beta^\alpha}{\sqrt{\pi}} \frac{x^{\alpha-1}}{\sqrt{1-\beta x}}$$ where $0<x<\beta$. So these are three terms all multiplied ...
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373 views

How to find $E(X|S)$ where $S=X+Y$ and $X$ and $Y$ are independent random variables with some density function $f$

So far I have this, but I am not sure if you are allowed to do this or if it is correct: I said since $X$ and $Y$ are independent and follow the same distribution, they must have the same ...
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41 views

Generating statistics from data samples, that are of snapshots.

My question is part math part computers but I need more of the math answer then a computer answer. I am going to be grabing snapshots of a moment of time on how a computer is performing. Because of ...
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455 views

How come i can't compute the expected value using the MGF for uniform distribution?

I thought that $M^r(0) = E[X^r]$, but for a uniform distribution the MGF is $\dfrac{e^{bt}-e^{at}}{(b-a)t}$, so there already is a singularity at $t=0$. So it would seem $M'(0) \neq E[X]$ Why is ...
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33k views

is it possible to calculate the standard deviation with a given mean and sample size?

I have been going in rounds with this problem... I may be thinking "complicated", any advice? I have the mean and total sample size (=number of data points) and I need to know what is the standard ...
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2answers
321 views

Is it correct to say that a sigma-algebra is generated by a collection of sets or a collection random variables ??

This really is a conceptual question. As I have come across articles saying that sigma-algebra is generated by collection of sets and articles that say sigma-algebra is generated by a collection of ...
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748 views

How to show the normal density integrates to 1?

How could you show that the normal density integrates to 1? $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1 $$
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CDF derivative of pdf? How does this work?

For a random variable $X$ if we have a pdf $f(x)$, then Continuous Random Variable is $$F(x) = \int_{-\infty}^{x}f(t)dt$$ Next $F'(x) = \frac{d}{dx}F(x)= f(x)$ I don't follow this, ...
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125 views

How do I find these bounds using Chebychev's inequality and the Central Limit Theorem?

Let $X$ have gamma distribution with parameters $\alpha=7$ and $\lambda=1$. Investigate the value of $F_X(10)$ using these methods: Find a lower bound using Chebychev's inequality. Approximate the ...
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597 views

posterior density for bayesian estimations

Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean $\lambda$. Based on previous experience in similar industrial plants, suppose ...
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85 views

Standard deviation problem

Let's say I have mean quality of 50 and standard dev of .1. And I the requirements for any product must be of quality 45 or higher. How do I calculate the chances that the quality of a product is ...
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0answers
55 views

How can I recover the weights of a Laspeyres index number?

I'm trying to recover the weights used to compute a Laspeyres quantity index. The index number's formula is: $$ Q=\frac{\sum_{i=1}^{N}{p_i^0 q_i^t}}{\sum_{i=1}^{N}{p_i^0 q_i^0}} $$ where in this ...
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230 views

Calculating the Odds of Victory in Risk

I am trying to write an odds calculator for risk that calculates the percentage chance of winning a combat between a number of given Attackers and given Defenders. The calculator will use basic risk ...
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108 views

Calculating the expected value of free tickets in a paytable?

Lets says I have a lottery game where the ticket costs $1 and has the following probability/prize distribution: 0.3 -> \$1 0.2 -> X 0.5 -> \$0 If X = \$1, then the expected value is: 0.3(\$1) + ...
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probability question need help

I got a question in statistics and probability, and i would like to have some kind of help in solving it. Question As you may have noticed, Dr. Mike says “right” in class, A LOT, and now Dr. Mike ...
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1answer
70 views

Finding a pdf using transformations

Let $X$ be a random variable that is $X \sim \mathrm{Unif}(0,1) = 1$. Use a transformation method to find the pdf of $U = X(1 - X)$. I tried solving for $X$ and I got $X = \dfrac{1 \pm ...
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851 views

Sample Range and order statistics?

Let a random sample of size $n$ from an exponential distribution $X_i \sim EXP(1)$. Give the pdf of (1) The sample range, $R = Y_n - Y_1$ (2) The first r order statistics The answers are supposed ...
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101 views

Predicting distribution based on drawings

A box contains 100 balls. Each ball has a number from 1 to 10. How many balls should I draw (ball is put back in box after drawing) to predict the number of balls for each number with 95% certainty. ...
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135 views

Application of Law of large numbers

Could you give me hints on solving the question below? Let $X_i$'s be iid r.v. Assume that they are mean zero ($EX_i$ = $0$) and they have finite variance. Consider $\bar{X_n} = \sum_{i = 1}^{n} ...
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How to test null hypothesis for spatial distribution?

I am given a sample from a spatial distribution $X$. For example, my variable $X$ is the number of certain diseases per city per capita. Null hypothesis is that variable $X$ does not depend on the ...
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761 views

Joint PDF and CDF two dimensional

Let $X$ and $Y$ have joint pdf $f(x,y) = 4e^{-2(x+y)}$ for $0 < x < \infty$, $0 < y < \infty$, and zero otherwise. (a) Find the CDF of $W = X + Y$ (b) Find the joint pdf of $U ...
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1k views

Sample variance of rolling a die 100 times

The textbook gives an example of testing a null hypothesis that rolling a die 100 times will give you a value of $6$, $\frac{1}{6}$ times. In the experiment, a die was rolled 100 times and 30 of them ...
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497 views

Pearson's Chi Squared / Cochran–Mantel–Haenszel test analog to N-way ANOVA

A test was given to two sets of students, CONTROL and EXPERIMENT, that had question A and question B. I want to know if students who got question A right were more likely to get question B right, and ...
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222 views

Numerical integration of binomial pdf with respect to a conditional probability

How do you numerically integrate the following loss function with respect to $\Phi(f_t)$, given $N_t=50$, $d_t=0$, $\rho=0.2$ and $\pi=0.01$? $P(D_t=d_t)=\bigl(\begin{smallmatrix} N_T \\ d_t ...
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2answers
323 views

Require brilliant resources to self teach.

I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
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158 views

Using GMM (Generalized Method of Moments) with 3rd and 4th order moments (skew and kurt)

I'd like to solve a GMM model with third-order and fourth-order moments for unknown (non-normal) distribution, i.e., get the asymptotic distribution of a vector theta consisting of the parameters that ...
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Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$

Does anyone have an example of two dependent random variables, that satisfy this relation? $E[f(X)f(Y)]=E[f(X)]E[f(Y)]$ for every function $f(t)$. Thanks. *edit: I still couldn't find an example. I ...
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Correlated Poisson Distribution

$X_1$ and $X_2$ are discrete stochastic variables. They can both be modeled by a Poisson process with arrival rates $\lambda_1$ and $\lambda_2$ respectively. $X_1$ and $X_2$ have a constant ...
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1answer
208 views

What sample size do I need to justify my suspicions?

I have been told that an event occurs about once in 50 times. In my experience, it is more like once in 40 or 45, which would not be insignificant (if correct). However because this event does not ...
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combinatorics: probability of picking elements

I have $6$ elements $$\{a, b, c, d, e, f \}$$ I close my eyes and randomly pick $5$ elements. What is the chance of getting $a$ and $b$ in those $5$ elements?
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1answer
40 views

Regression model for a shearing process

30 Widgets are randomly assigned to a shearing process. There are 3 such processes, each getting 10 widgets. The lengths of each widget are recorded before undergoing the shearing. The amount that ...
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177 views

Probability question on picking letters

5 letters: a, b, c, d, e I randomly pick 3 letters, whats the chance of having a and b in those 3 letters
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1k views

Hat Matrix Identities in Regression

I need to show that $\bar h= \sum{h_{ii}/n} = \operatorname{Tr}[H]/n = (p+1)/n$ Using the fact that $\operatorname{Tr}[AB]=\operatorname{Tr}[BA]$ and $H=X(X^TX)^{-1}X^T$. But I have no idea how to ...
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2answers
654 views

Obtaining cumulants using the characteristic function

If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as: $$\mu_n = ...
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2answers
234 views

Testing a hypothesis with significance level

Prove the hypothesis that the average content of containers of a particular lubricant is $10$ liters, if the contents of a random sample of $10$ containers are: \begin{array}{|c|c|c|c|c|} \hline ...
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115 views

Calculating variance, how to determine when to use 1/n or 1/(n-1)?

I'm learning multivariate analysis. I am asked to calculate covariance of $$X=\begin{pmatrix} 3&7 \\ 2&4 \\ 4&7 \end{pmatrix}$$ According to P8 of Applied Multivariate Statistical ...
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1answer
111 views

Is it possible to calculate/impute the sample size “N” from a given mean and standard deviation?

Can anyone provide with some advice? - thank you I am required to calculate the sample size for two groups, given the following data: Total sample N (group1+group2)=583 group 1: mean=8.35, SD1.07, ...
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5answers
24k views

How to intuitively understand eigenvalue and eigenvector?

I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman. Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to ...
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1answer
91 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
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63 views

Hypothesis testing - why $P(X \geq x_0 ; H_0)$ and not $P(X = x_0 ; H_0)$?

In a few of my statistics classes, I was taught that the procedure for carrying out a hypothesis test for a statistic $X$ with observed value $x_0$ is to determine the distribution under the null ...
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102 views

Mathematics or statistics?

Can you please tell me what is ((stochastic modelling and statistical analysis of spatio-temporal data)) related to? I mean Mathematics or statistics? Is it good subject for student who interested in ...
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How to show variance increases in an open system vs. a closed system?

Lets says I have 10 dogs and 1000 grams of premium dog food. I unleash said dogs in a room with said 1000 grams of dog food, weighing them before the experiment and then weighing them once more after ...
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69 views

Expection operator defined on colors

I am trying to implement a computer vision algorithm, but I'm having a problem with some notation used in a article. They define an image as a set of RGB colors with an index $\mathbb{z} = (z_1, z_2, ...
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120 views

Estimate number of distinct items

I have a large array of $n$ integers, some of which may be repeated, and I want to estimate how many distinct integers are in the array. Say the number of distinct integers is $N$. I can sample with ...
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1answer
174 views

sample distribution of sample variance for a random sample

I'm given that n=2 and a simple table showing this... x 0 1 5 p(x) .25 .25 .5 I found the sample distribution for the sample mean to be this... _ x 0 ...
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Estimated probability looks too large, what I am doing wrong? [duplicate]

A factory produces links for heavy metal chains. The research lab of the factory models the length (in cm) of a link by the random variable ${X}$, with expected value ${E(X) = 5}$ and variance ...