A routine question from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for self-study questions.

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What are the null and alternate hypotheses? Will you use a left, right, or two-tailed test?

Ben has a coin which he claims is weighted in a way so that when he flips it, heads appears more often than 50% of the time. He tries to prove it to you by flipping the coin 100 times, which ...
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1answer
11 views

How to evaluate a statistical test for fairness of a coin?

I suspect that a given coin is not fair, i.e that the probability of heads is not $\frac{1}{2}$. My question is not how we can prove the unfairness, but if we derive a test to check the fairness, How ...
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0answers
21 views

In confusion with a Bayesian statistical problem

I was learning some probability basics. I am stuck with a problem, that I need your help with in solving. An $e$-fair coin is a coin with probability of head $(\theta)$ in interval ...
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0answers
37 views

Basic problem in Bayesian inference

I have questions with the following Bayesian inference problem I found in the book by Bertsekas & Tsitsiklis (Introduction to Probability 2nd ed.). Problem is as follows (P.445, Problem 2): ...
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1answer
55 views

sum and product rules of probability

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
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24 views

Hyperplane data points

Would a single data point in the hyperplane (see below) correspond to a single cell in the data matrix or an entire row?
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1answer
39 views

Optimal Allocation

A campus population of size N=9000 is to be surveyed by a stratified sample for the prevalence of a certain disease based upon three strata of respective sizes $N_h$ = 1000, 3000, and 5000 for h = 1, ...
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2answers
88 views

Hat matrix,$H=X(X^{\prime}X )^{-1}X^{\prime}$

What is the importance of hat matrix, $H=X(X^{\prime}X )^{-1}X^{\prime}$ in regression analysis? Is it only for easier calculation ?
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1answer
28 views

Getting all zero correlations,$\rho_{ij}=\frac{\mathbb cov(e_i,e_j)}{(V(e_i)V(e_j))^{1/2}}$

Consider the general regression model $$Y=X\beta+\epsilon$$ where, $Y$ is an $(n\times 1)$ vector of observations, $X$ is an $(n\times p)$ matrix of known form, $\beta$ is a $(p\times 1)$ vector ...
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1answer
27 views

Uncertainty consistency?

The problem statement, all variables and given/known data I am given a set of x and y values x: (1,2,3,etc.) y: (1.2,2.2,3.1,etc.) with a given uncertainty and am asked a) find the best fit b) ...
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1answer
43 views

How to do multivariate regression in R?

I need to preform a multivariate normal regression in R. The question is: Let $Y_1$, $Y_2$, and $Y_3$ follows multivariate normal distribution. What is the conditional of $Y_3$ given ...
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1answer
45 views

Most efficient way to check answers to exercises when learning from a statistics textbook?

There's a community wiki at: Free statistical textbooks that has a nice list of freely available textbooks on statistics. Many of the textbooks that people suggest have exercises at the end of each ...
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1answer
17 views

Caculate Type II error for Binomial distribution

Basically this is a question if I got the general idea behind the Type II Error right. I’ll set up a little reproducible example: Given $p_0=0.6$ with an $\alpha$-level of $\alpha=0.1$. As Binomial ...
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0answers
23 views

Exponential distribution qq plot homework question

For an exponential qq plot, we fix the theoretical distribution to have λ=1. In order to compare the plotted points with the line y=x, what kind of rescaling should be applied to the data and why? ...
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1answer
23 views

Confidence Interval and Signifcance test Question

I have done the first part of the question already I just need help on the second part. You read in a U.S Census Bureau that a 99% confidence interval for the mean income in 2005 of American ...
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0answers
17 views

Statistics:Power Question Update [duplicate]

I have already asked this question but I did not initially give out all the details on the question, like the question being a multiple choice question. You manufacture and sell a iron rod whose ...
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1answer
60 views

Statistics: Power question

I have already answered this question I just like some feedback on am I right or wrong? You manufacture and sell a iron rod whose electrical conductivity is supposed to be 10.1. You plan to make ...
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26 views

Determining which t-test is right?

I have already solved this question, but I need some feedback. A new method for measuring blood pressure in laboratory mice using radiotelemetry has been proposed. How does this compare with the ...
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2answers
77 views

Distribution function of maximum of n iid standard uniform random variables where n is poisson distributed

I am studying probability theory on my own and am trying to work the following problem in the book - Let $X_1, X_2, . . .$ be independent, $U(0, 1)$-distributed random variables, and let $Nm \in ...
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1answer
102 views

Question with MLE

I'm having some problems with this question, and was hoping someone here could help. Let $X_1,\ldots,X_2$ be $n$ determinations of a physical constant $\theta$. Consider the model $X_i = ...
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1answer
100 views

What does it mean “being normally distributed”

There is an exercise which is used to illustrate how normal distribution works. The exercise starts by saying "Suppose scores on an IQ test are normally distributed..."; What does it mean for the ...
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1answer
129 views

What is the two-sample CDF of of $D^{+}$ and $D^{-}$ from the one-sided Kolmogorov-Smirnov Test?

I am trying to understand how to obtain $p$-values for the one-sided Kolmogorov-Smirnov test, and am struggling to find CDFs for $D^{+}_{n_{1},n_{2}}$ and $D^{-}_{n_{1},n_{2}}$ in the two-sample case. ...
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1answer
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Plain english explanation of the Rayleigh distribution?

I need to understand the Rayleigh distribution for a homework assignment in computer networks. Unfortunately, I lack the background knowledge in the field of statistics and probability theory to ...
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1answer
41 views

Number of distinct bootstrap samples

For $n$ distinct observations, there are ${2n - 1 \choose n-1}$ distinct bootstrap (re)samples. Could someone please provide a simple explanation? I found ...
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1answer
39 views

How to show linear model corresponds to exponential family

I am confronted with the exercise below. I have no given solution, so I hope someone can tell me whether my solution is right or wrong. I want to show the deterministic linear model $\quad ...
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1answer
134 views

What subject is more useful for statistics students [closed]

I am deciding which of the following two subjects to take as one elective in the coming term. Complex Analysis Theory of Partial Differential Equations Ideally I'd like to take both. However, I ...
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1answer
25 views

PDF of a sum using convolution

Let $X$ and $Y$ be independent, continuous random variables. The PDF of a sum $W=X+Y$ is given by the convolution formula $$ f_{X+Y} (w) = \int_{-\infty}^{\infty} f_X(x) f_Y(w-x)\,\,dx$$ In this ...
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1answer
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Tricky question about MLE

$X_1, \ldots, X_n$ iid ~ Pois($\lambda$). Suppose, you don't know the value of each $X_i$, but you know if $X_i = 0$ or not for every i. Find MLE for $\lambda$. Does MLE always exist? I ...
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Unable to understand fuzzy logic example

I shall be grateful for an explanation of how to apply fuzzy logic based reasoning for the following example: I have 3 crisp sets of features = color, shape and texture. The objects are red ball, red ...
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1answer
45 views

Which of these points in this plot has the highest leverage and why?

I am studying the definition of leverage, and I understand it in terms of formulas. However, if I would have a plot like this for instance, how could I see which of these points has the highest ...
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25 views

Law of iterated expectations - an small exercise

From edX MIT probability course - Widgets and Crates: Let $X_i$ be the number of widgets in a particular box $i$. Let $N$ be the number of boxes in a crate. Assume $X$ and $N$ are independent, ...
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31 views

Maximum likelihood estimator for variance in two linear models

I am learning MLE's at my inference class and this is a problem I came accross. Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha ...
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1answer
95 views

Asymptotically unbiased estimator using MLE

I am learning Maximum likelihood estimators for a inference class. And this is a problem I came across. Let $X_1,X_2,X_3,\ldots, X_n$ be a random sample with p.m.f $$p(X)=\theta(1-\theta)^x; ...
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0answers
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UMVUE for normal distribution $\sigma$

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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Maximum Likelihood estimators in reation to linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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2answers
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Pivotal to estimate lambda of a exponential

I am studying interval estimation by the method of pivotal quantities. Let $X_1,X_2,...,X_n$ be a random sample from a p.d.f $f(x;\lambda)=\lambda e^{-\lambda x}, x>0,\lambda >0$. I have to ...
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1answer
87 views

Advantage of Maximum Likelihood estimation [closed]

We can estimate unknown parameters by Least square, Least Mean Square, Blue estimators and Maximum Likelihood estimation. Q1: What is the advantage of MLE over others and when should MLE be chosen? ...
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2answers
103 views

Is a one sample t test the right approach for this question?

This is an example from my book that I'm trying to figure out but I'm totally lost. Can someone complete this step by step so I understand? A new meditation technique based on making barnyard ...
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1answer
54 views

Weighted least squares

Consider the estimator $b_1=\frac{\sum y_i}{\sum x_i}$. Suppose that $y_i = \beta x_i + \epsilon_i$, $E[\epsilon_i]=0$, $E[\epsilon_i \epsilon_j] (i \neq j)$ and $E[\epsilon_i^2]=\sigma_i^2$. Find ...
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1answer
184 views

Probability mass function of the sample mean

Suppose that $X_1,X_2,\ldots,X_n$ is a random sample from a distribution with probability function: $$p_X(x)=\begin{cases} 1/2 &x=-1,1 \\ 0 & \text{otherwise} \end{cases}$$ Now if we define ...
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0answers
16 views

categorical variables in regression analysis and interaction terms

I am learning multiple regression with categorical variables and in a book I came across this problem. For yield of corn suppose there are two factors affecting, nitrogen level and depth of ...
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18 views

Interaction terms in regression-theory

I am learning categorical variables and interaction terms in the regression course and there few things I don't understand. Suppose a salary data set with predictors experience $X$ in years ...
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1answer
103 views

exercise in probability

Consider the situation where n items are to be partitioned into $k < n$ distinct subsets. The multinomial coefficients. provide the number of distinct partitions where $n_1$ items are in group ...
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22 views

Poisson autocorrelation function

I want to draw an autocorrelation function, call it, AC(tau) where tau is the offset in the autocorrelatin. The vector I'm feeding into the AC is the spike train of a neuron whose action potential ...
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0answers
12 views

Time interval histogram (tih), Fano factor, coefficient of variance

I'm given this: A pacemaker neuron fires 10 spikes/s in a fairly regular manner, with interspike intervals (ISIs) that are distributed according to $\mathcal N(100, 5)$. I'm asked to draw the time ...
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Variance of sample mean in a binomial distribution

I'm trying to derive that the variance of phat = pq/n. I'm unfortunately not getting that. Please tell me where I'm going wrong... ...
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1answer
184 views

Simulate a random number of sums of random variables?

I'm trying to solve an exercise (not homework, and it does not come with solutions) that asks to provide an algorithm to simulate a probability $P(Y<y)$ and the expected value $E[Y]$ of a random ...
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1answer
82 views

Whether lead content readings follow a Normal distribution or not

Studies indicated that drinking water supplied by some old lead-lined city piping systems contain harmful levels of lead. It appears that the distribution of lead content readings for individual ...
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1answer
28 views

Showing characteristic function is infinitely differentiable

Suppose that X is a real-valued random variable. Suppose that there exists a constant M > 0 such that the support of X lies entirely in the interval $[-M,M]$. Let $\phi$ denote the characteristic ...
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1answer
39 views

A question on discriminant analysis- Linear discriminant function

Above is part of an examination paper. I am not sure how to understand this SAS output. Especially what is there in the last table which looks to me like two discriminant functions. Can someone ...