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Questions tagged [self-study]

A routine exercise designed to test one's knowledge; often 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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4 votes
1 answer
231 views

Bayes net probability question

I've made this Bayes net based on a problem and I'm trying to find the probability of W but I'm stuck. I know I probably have to use Bayes theorem backwards through to find $P(W)$, but I'm not sure ...
2 votes
3 answers
458 views

Poisson Distribution vs Binomial Distribution

I have a specific problem here. It is stated as: "Assume that 30% of students in a university take public transportation daily to commute to their college. Suppose 10 of the students are randomly ...
1 vote
1 answer
284 views

How to introduce the error bars in text?

I want explictly to state in my written text that the error bars in the current plot and all of the following ones represent the 95% percent of the calculated mean. However, the metric changes in the ...
0 votes
0 answers
23 views

Is there an additive property for hypergeometric random variables?

I know that for binomial and negative binomial RVs there is an additive property where if $X_1\sim bin(a, p)$ and $X_2\sim bin(b, p)$ then $X_1+X2 \sim bin(a+b, p)$ if $Y_1\sim NB(c, p)$ and $Y_1\sim ...
6 votes
2 answers
6k 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 $N_m \in Po(m)$...
3 votes
1 answer
462 views

UMVUE for $g(\theta)=\theta^2$ of Poisson random variables

Let $X_1,...X_n$i.i.d.~$Pois(\theta)$ with unknown $\theta>0$, I want to find the UMVUE for $g(\theta)=\theta^2$. I know that $T(x)=\Sigma_{i=1}^{n}x_i$ is complete and sufficient for $\theta>0$....
6 votes
1 answer
3k views

Likelihood Ratio Test Equivalent with $t$ test: Difference of Two Means from Constant Variance Normal Distributions

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Suppose that independent random samples of sizes $n_1$ and $n_2$ are to be selected from normal ...
5 votes
2 answers
930 views

Wald statistic with known mean and unknown variance in a Normal Distribution

I'm working from the Casella-Berger book and I've run across this problem, I've managed to answer part (a) and most of part (b); however, upon looking in the solutions manual, I'm not able to get the ...
19 votes
3 answers
2k views

Proof/derivation for false discovery rate in Benjamini-Hochberg procedure

The Benjamini-Hochberg procedure is a method that corrects for multiple comparisons and has a false discovery rate (FDR) equal to $\alpha$. Or is it the family wise error rate, FWER? I am a bit ...
3 votes
1 answer
239 views

What test to use after a PCA?

Need some help answering a prac exam question "An ecologist is interested in the morphological adaptations of goannas to their microhabitat. She wishes to determine if goanna shape differs for ...
1 vote
1 answer
743 views

What is the relationship between R-squared and MS(Res)?

My textbook (Applied Regression Analysis: A Research Tool by Rawlings, Pantula, and Dickey) asks me to "Show algebraically the relationship between R-squared and MS(Res)", but I don't even ...
3 votes
1 answer
641 views

Inference in Dirichlet process mixtures via collapsed Gibbs sampling

I need to cluster some data $\{x1, \ldots, x_n\}$ through a Dirichlet process mixture model. Consider the following Dirichlet process mixture model, in which the base measure is a $NIW(\mu_0, \...
0 votes
0 answers
28 views

Exam P Sample Problems # 30

The question: A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a ...
71 votes
6 answers
231k views

Find expected value using CDF

I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing. ...
1 vote
1 answer
245 views

How to calculate dimension of weight matrix for a vanilla RNN

Sorry if this is kind of a dumb question but I'm taking the NLP course from Andrew Ng on Coursera and I don't understand how to arrive at the correct answer for a question pertaining to calculating ...
2 votes
3 answers
149 views

Conditional probability, who's right?

i need help with a math problem. Me, and at least two teachers have all done it and gotten different results, so now I'm asking the wise people of the internet for help settling the debate. Here's the ...
26 votes
2 answers
28k views

How to understand that MLE of Variance is biased in a Gaussian distribution?

I'm reading PRML and I don't understand the picture. Could you please give some hints to understand the picture and why the MLE of variance in a Gaussian distribution is biased? formula 1.55: $$ \mu_{...
8 votes
1 answer
6k views

K-means as a limit case of EM algorithm for Gaussian mixtures with covariances $\epsilon^2 I$ going to $0$

My goal is to see that K-means algorithm is in fact Expectation-Maximization algorithm for Gaussian mixtures in which all components have covariance $\sigma^2 I$ in the limit as $\lim_{\sigma \to 0}$. ...
4 votes
1 answer
214 views

Exercise about Order statistics from uniform distribution

I'm trying to solve an exercise about order statistics. The exercise is the following: Let $U_{(1)}< \ldots <U_{(n)}$ be the order statistics from Uniform distribution U(0,1). Show that $(-\log[...
8 votes
2 answers
310 views

Conditional Expectation of Product of Normals given a Linear Combination

I am tasked with solving a question for a qualifying exam, but I am a little lost about this question. Let $\eta$ and $\xi$ be two independent standard Gaussian random variables. Find $\mathbb{E}(\xi\...
2 votes
1 answer
165 views

I need to prove that $\hat\theta=\max\{X_1,...,X_n\}$ is a mean square consistent estimator for $\theta$

Let $X_1,...,X_n$ a i.i.d from a population with distribution $U[0,\theta]$, i.e., $f_{X_i}(x)=\frac{1}{\theta}g_{[0,\theta]}(x)$, for $i=1, \ldots, n$ where \begin{align} g_{[0,\theta]}(x) = \begin{...
1 vote
0 answers
46 views

What is the distribution of the unbiased estimator of variance for normally distributed variables?

I must be making some mistake in my derivation of the distribution of the unbiased variance estimator for i.i.d. $X_{i} \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$. We have $\bar{X} =\frac{1}{n}\sum\...
3 votes
1 answer
239 views

Calculation of residual standard deviation and r-squared

Related questions here and here (but not answered that satisfactorily in my view). From Gelman and Hill, Q3.2: Suppose that, for a certain population, we can predict log earnings from log height as ...
14 votes
1 answer
6k views

Distribution of a quadratic form, non-central chi-squared distribution

Definition. Suppose $\mathbf{y} \sim \mathcal{N}(\boldsymbol{\mu}, I_{n \times n})$. Then $$w = \mathbf{y}^{T}\mathbf{y} = \|\mathbf{y}\|^2 \sim \chi^{2}_{n}\left(\theta = \|\boldsymbol{\mu}\|^2/...
1 vote
1 answer
322 views

Conditional Expectation(s) of Multiple Random Variables

I am having a bit of trouble with one of the problems I am working right now. The problem is: "What is $E\left \{ E\left \{ E\left \{ Z \mid X , Y \right \} \right \} \right \}$ ?" During an ...
0 votes
0 answers
38 views

Likelihood function for data with random censoring

The following is from Klein and Moeschberger, p. 76. Let $(T,\delta)$ be a tuple with $T = \min(X,C_r)$ and $\delta = 0$ if the lifetime X is censored and $\delta = 1$ if it is not; $C_r$ denotes the ...
3 votes
1 answer
342 views

Another EM-algorithm problem

I have the following problem: I have a random vector $y$ which has length $l$. The first $z$ bits come from a Bernoulli random variable with parameter $\theta_1$ and the next $l-z$ come from a ...
2 votes
1 answer
125 views

Show that $R_{n}^{2}-n$ and $(-1)^{n} \cos(\pi R_{n}) $ are $\mathcal F_{n}$-martingales

Let $X_{i}, i\ge 1$, be i.i.d. random variables defined on a probability space $(\Omega, \mathcal F,P)$ such that $P(X_{i}=1)=P(X_{i}=-1)=\frac{1}{2}$. Consider the filtration $\mathcal F_{n}=\sigma(...
3 votes
1 answer
372 views

Is this a linear or non linear model, and why?

A model $Y=(\beta_0+\beta_1x)^{-1}+\epsilon$, where $\epsilon \sim N(0,\sigma^2)$ is to be fitted to the data $(x_1,Y_1), (x_2,Y_2), \dots (x_n,Y_n)$. Is this a linear or non linear model, and why? ...
2 votes
0 answers
48 views

What to do in Box-Jenkins framework when time series has deterministic trend and seasonality?

I'm self-studying time series and I'm puzzled by apparent lack of consistency between : the "classical" decomposition of time-series and the Box-Jenkins methodology. Concerning the ...
0 votes
1 answer
61 views

How to solve this "easy" probability problem

I'm struggling on how can I solve the following problem. I tried with a lot of calculus without success, but i think that the way to solve is: $$P(\text{at least one failure})=1−P(\text{no failures in ...
1 vote
1 answer
84 views

Understanding the Logistic regression formula

Logistic regression aims at transforming the linear regression formula and fitting the s curve or logistic function to a particular dataset in order to calculate the probability of a categorical ...
0 votes
0 answers
35 views

Sufficient Statistic for Truncated Normal

I am doing exercise 3.18 of "The Bayesian Choice": Give a sufficient statistic associated with a sample $x_1,...,x_n$ from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x ...
3 votes
3 answers
772 views

How to prove any one-to-one function of minimal sufficient statistic is minimal sufficient?

So I want to prove that any one-to-one function of minimal sufficient statistic is also minimal sufficient. Here is my proof: Let $T$ be a minimal sufficient statistic and $f$ is a one-to-one function ...
1 vote
1 answer
39 views

Unsure about assumptions of linear model with time series variables, spurious regression and periodic patterns

Background I'm learning about time series in context of linear regression. The goal of this question is to understand how seasonality of either X or Y can affect the model. Linear model assumptions $...
0 votes
0 answers
22 views

Computing Bayesian model averaged posteriors

The Bayesian model averaged posterior predictive distribution for new data $\tilde{y}$ given training data $y$, across a set of $M$ models $\mathcal{D} = \{D_{1}, ..., D_{M}\}$, is defined as: \begin{...
3 votes
2 answers
258 views

Let $X_1, X_2, X_3 \sim \textrm{Expo}(1)$ distribution. How to compute $\mathrm{P}(X_1 + X_2 \leq rX_3)$?

So let $X_1, X_2, X_3$ be random samples from the $\textrm{Expo}(1)$ distribution. How do I set up the computation for $\mathrm{P}(X_1 + X_2 \leq rX_3 | \sum_{i=1}^n X_i = t)$ where $r, t > 0$? ...
2 votes
1 answer
40 views

Bayes Theorem and Total Probability

I'm back for more help....I'm terrible with conditional probability. Here's a problem from an assignment in a stats course I'm taking: A U.S. Army Corps of Engineers (USACE) study focused on DDT ...
2 votes
1 answer
747 views

Conditional Probability on Disease

A man living in a country where only 1 out of 1000 people has the virus A. There is a test available that gives a positive result 5% of the time when the patient does not have virus A and a negative ...
3 votes
2 answers
308 views

How to compute the joint distribution of transformed variables using the Jacobian?

$X$ and $Y$ are independent continuous random variables and have the same distribution $F_x(t) = 1 - (1/t)$ for $t \gt 1$. We define two new variables $W$ and $Z,$ where $W = \min(X,Y)$ and $Z = \...
7 votes
3 answers
1k views

What is the zero-truncated Poisson distribution used for? And how are 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 are $$\operatorname E(Y) = \frac{\lambda}{1-e^{-\...
2 votes
2 answers
324 views

How to estimate maximum liklihood of a custom log likelihood function?

I am not very familiar with maximum likelihood estimation. But I would like to test the null hypothesis $\mu = 0, \sigma = 1, \rho = 0$ by estimating the following model: $$z_t - \mu = \rho(z_{t-1} - ...
1 vote
0 answers
21 views

What is the formula to compute the height of significance line on autocorrelation plot?

Autocorrelation of a time series can be plotted in R with use of acf function. For example: acf(ldeaths) # built-in series I ...
2 votes
1 answer
281 views

Same odds ratio for both rows and columns

I'm getting stuck on how to run the odds ratio test for these two different questions: Novice clinicians are more likely to use validated assessments than other assessments Novice clinicians are ...
0 votes
1 answer
4k views

What is the posterior distribution of θ? Is the Gamma a conjugate prior for an exponential likelihood?

A manufacturer is interested in the time to failure of his batteries. Suppose the time to failure of the batteries has an exponential distribution: $$p(x│\theta)=\theta\exp(-\theta x)$$ Note that the ...
4 votes
2 answers
639 views

If Y has an exponential family distribution show that $E(\frac{\partial L}{\partial \theta}) = 0$

I'm working in a self study fashion preparing for a course I'm going to take this semester in generalised linear models. The question is, given that the Y random variable belongs to the exponential ...
4 votes
2 answers
206 views

Chi-Square Test Problem

a. 56 cells (87.5%) have expected count less than 5. The minimum expected count is .02. Df: 49 P Value: .049 Is this result significant or I may select any other test?. Please guide me.
2 votes
1 answer
132 views

2^3 Factorial design in Yates' notation

Consider a $2^3$ factorial design lay out in 2 blocks, each of size 4, as follows Block I {1, a, b, c} Block II {ab, ac, bc, abc} Here, the treatments combinations are written in Yates’ notation. ...
1 vote
1 answer
74 views

Randomised Block Design

Suppose the response of three different treatments, $A$, $B$ and $C$ are measured in two different hospitals of a country. The data are given below. My question is: so far I understand it is a ...
0 votes
1 answer
97 views

Proving the average of sum of i.i.d cauchy is not a consistent estimator of location parameter

Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$ To prove that it is inconsistent, consider the characteristic function of $X_i$ and $\bar{X}$, which ...

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