Questions tagged [self-study]

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

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2answers
109 views

MCMC algorithm going wrong [closed]

Given this integral \begin{equation} \int_0^\infty \chi_{[1,2]}(x)\Gamma(C,x)\left|\cos(R x)\right| \, dx \end{equation} where $\chi_{[1,2]}=\begin{cases}1, & x \in [1,2] \\ 0, & x\not \in [...
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0answers
12 views

Different classification loss for K-nearest neighbours

Suppose we have a general classification loss instead of a 0-1 loss. How can we modify k-NN to accommodate such a loss function? I thought about using a weighted loss matrix where $L(i,j)=0$ when $i=...
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1answer
62 views

Finding the characteristic function of a simple linear PDF

X is a random variable with a pdf of $ f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}$ I tried finding the characteristic function of this but ended up with ...
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0answers
15 views

How do I find the probability of interference of a shaft and a bearing given their nominal diameter values and their standard deviations?

The exact question is as follows: An assembling of shaft and bearing is made out of shaft manufactured to a specification of 30.00 ± 0.09 mm and bearings are manufactured to a specification of 30.10 ±...
1
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1answer
54 views

Why should we study copulas? [closed]

I am new to the study of copulas and I would like that someone could provide some examples where they are applied, their usefulness and so on. Any help is appreciated. Thanks in advance.
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0answers
42 views

What receptive field do we have after stacking $n \times n$ CONV layers with kernel size $k \times k$?

What receptive field do we have after stacking $n \times n$ convolutional layers with kernel size $k \times k$ and stride $1$? Layers numeration starts with $1$. The resulting receptive field will be ...
6
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1answer
91 views

Maximum Likelihood Estimator of $P(Y_1=1)$ where $Y_i=1$ if $X_i>0$ and $0$ otherwise, given $X_1,\dots,X_n\sim N(\theta,1)$

This is part(a) of exercise 6 of Chapter 9 from Wasserman's All of Statistics. Let $X_1,\dots,X_n\sim N(\theta,1)$. Define $Y_i=\begin{cases} 1 &\text{ if }X_i>0 \\ 0 &\text{ if }X_i\le 0....
0
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0answers
20 views

Time Series: ACF and PACF plot, how to tell what's the best model by looking at the plot?

The question ask me "The equation of the model you think is most appropriate, given the plots. Justify your choice of model. " the ACF plot looks like cut off after the first lag. And I'm not too sure ...
1
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1answer
40 views

Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$ P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no ...
0
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1answer
34 views

Unbiased variance question [closed]

A researcher is testing if a new swimming technique is more effective. She knows the average 50m time of swimmers in her club using the old technique is 35 seconds. After training 12 swimmers with the ...
1
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1answer
42 views

Proving the Accepted Samples from Rejection Sampling follows our Posterior Distribution

I get confused how the author gets to line $(1)$ using the indicator functions. If someone can explain this to me or give me a hint that would be greatly appreciated. Thanks! Define the following: ...
3
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1answer
95 views

Neyman-Pearson hypothesis testing with uniform random variables

This question is quite simple. We have a random sample $X_1, X_2, ..., X_n$ from $U(\theta, \theta + 1)$ and we want to test $H_0: \theta=0$ vs $H_0: \theta=\theta_1$ for some $0 < \theta_1 < 1$....
4
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3answers
183 views

Proving transformations of two independent chi-squared random variables is equivalent to a Beta distribution

I came across the following in some old class notes of mine: if $\chi_{v_{1}}^{2}$ is independent of $\chi_{v_{2}}^{2}$ then $\frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}}\backsim ...
2
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1answer
55 views

Moment Generating Function of Beta ( Taylor series)

Suppose X is a random variable with a Beta ( a =$\frac{1}{2}$ , b=1) distribution and x in (0,1) Then the moment generating function is calculated as below $ M_X(t) $ = $\mathbb{E}[e^{tX}]$ =$ \...
0
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0answers
16 views

How to determine hypothesis and evidence in Bayes theorem

I am new to Bayes theorem, and bit confused how to identify evidence and hypothesis in the theorem.For e.g.I have this problem - in a company 10% of laptops fail within their warranty period. 5% of ...
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1answer
22 views

Integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$

I'm trying to integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$ using the fact that the integral of any normal PDF is 1. But I'm having trouble completing the ...
3
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1answer
68 views

Why normality assumption on linear model implies equivalence between least square estimation and maximum likelihood estimation?

Consider the following excerpt from the Alan Agresti's book on generalized linear models: "Having formed a model matrix $\textbf{X}$ and observed $\textbf{y}$, how do we obtain parameter estimates $\...
0
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1answer
20 views

How to determine N of LOOCV

In my textbook, it says that LOOCV is where $K=N$, but how do I find the value of $N$? Is it just $K-1$?
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1answer
21 views

Calculate $\mathbb{P}[Y=y|X=x]$ where $X$=# claims reported diring firs year, $Y$=# claims that will eventually be reported

A property-casualty insurance company issues automobile policies on a calendar year basis only. Let $X$ be a random variable representing the number of accident claims reported during calendar year ...
0
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0answers
18 views

Unbiased estimator ( Integral Issue)

$X_1, ... , X_n$ are iid with pdf $f(x|\theta) = e^{-(x-\theta))}I_{(\theta, \infty)}(x)$ it is easy to find the sufficient statistic which is $X_{(1)}$ $E_{\theta}[g(X_{(1)})]=$ $\theta$ (...
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0answers
40 views

Trouble with copulas: how do we justify its definition?

A bivariate function $C(u,v)$ that maps $[0,1]^{2}$ to $[0,1]$ is a copula if it satisfies the following two conditions: (i) Boundary conditions: \begin{align*} C(u,0) = 0\\ C(0,v) = 0\\ C(u,1) = u\\ ...
1
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1answer
48 views

Unbiased Estimator based on Sufficient Statistic

suppose $X_1, ... , X_n$ are iid with pdf $f(x|\beta) = e^{-(x-\beta))}I_{(\beta, \infty)}(x)$ and the pdf of ( the smallest order statistic) $X_{(1)}$ is given by $f_{X_1}(x)$ = n $ *$ $e^{n(\...
2
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2answers
113 views

How can I identify these time series processes? (AR/MA/ARIMA/random walk with drift)

I don't understand how one would identify the stochastic process of the following models, if they are AR or MA or ARIMA etc. Consider the following models estimated over a sample $t = 1, 2, \dots,T$...
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0answers
8 views

Variance of interblock estimator in balanced incomplete block design

I am working in a balanced incomplete block design, whose model is as follows: $$y_{ij} = \mu + \tau_i + \beta_j + \epsilon_{ij}, i = 1\ldots a, j = 1 \ldots b$$ where $y_{ij}$ is the value of the ...
5
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2answers
254 views

Distribution normality check

I can not solve the problem from my homework. We conducted two experiments. In the first, there were 400 patients, and in the second, 250. In these experiments, the effects of various drugs ...
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0answers
18 views

One-sided confidence interval for correlation coefficient

Consider this question, Let $(X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n)$ be independent and identically distributed pairs of random variables with $E(X_1) = E(Y_1), Var(X_1)= Var(Y_1) = 1$, and $Cov(X_1,...
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1answer
21 views

First two conditions of the Ols estimate

this was a question in my previous test. The answer given to this is 'd'. But from what I know the option given in 'b' are the two first order conditions of calculating an ordinary least squared ...
2
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0answers
22 views

Understanding the solution to a problem about a homogeneous Poisson process

This is probably easy, but right now I can't figure it out, so bear with me. The question is: Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=...
0
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1answer
44 views

Compute $E(X_1|X_1+X_2)$ $X_1, X_2$ both iid $Exponential(1)$

I recently stumbled across this question on CV: Conditional expectation conditional on exponential random variable And really liked the answer provided by @Rush, but I wanted to try to compute this ...
1
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0answers
59 views

Karlin Rubin Theorem UMP (Uniformly most powerful test ) Is it correct?

Suppose $X_1, X_2, X_3,\ldots, X_n$ are i.i.d. random variables with a common Poisson$(\lambda)$ distribution. $$X=(X_1, X_2, X_3,\ldots, X_n)$$ and $g(λ)=\lambda(1 - e^{-λ})$ , with $(λ>0)$ ...
3
votes
3answers
113 views

Maximum likelihood estimator of $n$ when $X \sim \mathrm{Bin}(n,p)$

Given a random variable $X\sim Bin(n,p)$, where $p$ is known $p\in (0,1)$ , $n$ is an unknown positive integer and $x\in\{0,1,2,....n\}$, what is the maximum likelihood estimator of $n$? I ...
1
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0answers
24 views

Fixed Effects model

I am trying to understand the fixed effects model. $ Y = D\alpha + X\beta + \epsilon$ Where $\alpha$ is the fixed time invariant effect. If we model this using Frisch Waugh lovell partitioning ...
1
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0answers
24 views

Conditional Expectation of Poisson

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common pmf poisson(λ) (t = a value) How would you calculate the below without using intuition (I would appreciate if you ...
1
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1answer
40 views

I have derived the Mean and Variance of a truncated Poisson distribution. Does this show under-, equi-, or overdispersion?

The density looks like this: $P(Y=y) = \frac{e^{-\lambda} \lambda^y}{y!(1-e^{-\lambda})}$. I derived the mean and variance and got this: $$\operatorname E(Y) = \frac{\lambda}{1-e^{-\lambda}}$$ $$ \...
0
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1answer
44 views

How do we find the maximum likelihood estimate of $\mu$ from $\mathcal{N}(\mu,\sigma^{2})$ through the Newton-Raphson method if $\sigma^{2}$ is known?

Here it is the problem: I am supposed to obtain the maximum likelihood estimate of the mean for some normal distribution $\mathcal{N}(\mu,\sigma^{2})$ where $\sigma^{2}$ is known (let it be $\sigma = ...
1
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1answer
33 views

Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
1
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1answer
39 views

EM Algorithm for Poisson Gamma

I would like to check if I have done this question right. I am trying to derive the EM algorithm for $\mu$ in the following distributions: $$f(Y|Z) = \frac{z^y}{y!} e^{-z} $$ $$f(Z) = \frac{\theta^\...
1
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0answers
23 views

hypothesis testing doubt

I have the OLS regression model: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 + \epsilon$ I want to check the hypothesis: Ho : $\beta_2*\beta_3$ = 1 Will I use the Delta ...
0
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0answers
25 views

OLS Heteroskedasticity correction

I have a data and was trying to correct for heteroskedasticity (which is significant as per Breush Pagan test). However, after using the robust command in stata, my standard errors of almost all the ...
0
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0answers
25 views

Difficulty with problem 3.11(d) in An Introduction to Statistical Learning (ISLR) (simplifying an equation)

I'm working through An Introduction to Statistical Learning and am getting stuck on question 3.11(d) from the third chapter. In short: For the regression of Y onto X without an intercept, the t-...
2
votes
2answers
67 views

What is the zero-truncated Poisson distribution used for? And how is the mean and variance derived?

I know that the density looks like this: $P(Y=y) = \frac{e^{-\lambda} \lambda^y}{y!(1-e^{-\lambda})}$ and from wikipedia that the mean and variance like this: $$\operatorname E(Y) = \frac{\lambda}{1-...
1
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1answer
40 views

Literature / Books on Linear Models, Generalized Linear Models and Linear Mixed Models

As the title suggests, I'm looking for book recommendations on Linear Models, Generalized Linear Models and Linear Mixed Models. The book should give a good overview on the intuition behind ...
5
votes
2answers
122 views

Distribution of $XY$ when $(X,Y) \sim BVN(0,0,1,1,\rho)$

The question is pretty much in the title, I need to find an approximate distribution of $XY$ when $(X,Y)$ follow a Bivariate Normal Distribution where $X$ and $Y$ are each $N(0,1)$ distributed and $...
0
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0answers
9 views

F test, Model selection test

How to compare which of the following two models is a better fit: M1 $y$ = $\alpha$$X$ + $\epsilon$ M2 $ln(y)$ = $\alpha$$X$ + $\epsilon$ Can we run a test statistic to compare the two models? ...
1
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0answers
30 views

Maximum likelihood estimation versus a given interval

I have been solving some exercises on Maximum Likelihood estimation and I got across this one- It is known that the proportion of smokers (p) in a population lies in the interval $[1/3, 2/3]$. In a ...
0
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0answers
21 views

Self-study: Which option is NOT a prediction of the central limit theorem?

"All of the following are predictions of the Central Limit Theorem except: 1) The sample mean distribution will be approximately normally distributed if the sample size is large 2) The mean of the ...
-1
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1answer
78 views

Distribution of sums and differences of n correlated normal random variables

Suppose $x_1\sim\mathcal N(2,0.5),x_2\sim \mathcal N(2,3),$ and $x_3\sim \mathcal N(2.5,7)$ with correlations $\rho_{(1,2)}=0.3,\rho_{(1,3)}=0.1,$and $\rho_{(2,3)}=0.4.$ What is the distribution of ...
0
votes
1answer
47 views

Approximate distribution for sum of squares of standardized Poisson random variables

Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for $\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...
0
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0answers
15 views

What statistical property is absent in 2-sample hypothesis tests of 5 measurements?

Suppose we observe a random sample of five measurements: 10, 13, 15, 15, 17, from a normal distribution with unknown mean $\mu_1$ and unknown variance $\sigma_1^2$. A second random sample from another ...
2
votes
1answer
115 views

Proving the MVUE is the following

I am stuck on the following question and I was wondering if can get some help. Let $f(x;\theta) = g(\theta)h(x),\ a(\theta) \leqslant x \leqslant b(\theta)$ with $a(\theta)$ decreases and $b(\theta)$...