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3
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1answer
27 views

Why does the number of continuous uniform variables on (0,1) needed for their sum to exceed one have mean $e$?

Let us sum a stream of random variables, $X_i \overset{iid}\sim \mathcal{U}(0,1)$; let $Y$ be the number of terms we need for the total to exceed one, i.e. $Y$ is the smallest number such that $$X_1 ...
4
votes
0answers
11 views

Minimal sufficiency with indicator functions

The following theorem can be used to demonstrate that a statistic is minimal sufficient: Let $f(X|\theta)$ be the pmf or pdf of a sample X. Suppose $\exists$ a function $T(X)$ such that, for ...
0
votes
0answers
7 views

How do I compare the size and power of the tests?

Let $X_{1},\dots,X_{9}$ be a random sample from $N(\theta,1)$, where $-\infty<\theta<\infty$. Consider the following two tests for testing $H_{0}:\theta=2.5$ against $H_{1}:\theta=4$. Test1: ...
1
vote
1answer
14 views

Probability that 2 out of 3 friends win a raffle?

Let's say it's me and 2 other friends are at a contest. There are 10 total contestants. What is the probability that exactly two of us win (it's not ranked, so it's just a combination problem)? I'm ...
2
votes
1answer
44 views

Likelihood Ratio Test statistic for the exponential distribution

I need to test null hypothesis $\lambda = \frac12$ against the alternative hypothesis $\lambda \neq \frac12$ based on data $x_1, x_2, ..., x_n$ that follow the exponential distribution with parameter ...
0
votes
1answer
44 views

sampling distribution for N(0,1) samples

Here is an portion of my lecture notes from class, we are studying sampling distributions. I am confused on some of the examples that are showed in the attached picture. For the first example, ...
0
votes
0answers
7 views

how do you calculate nearest neighbor calculation times

I have this question: Suppose you are creating a website to help shoppers pick houses. Every time a user of your website visits the webpage for a specific house, you want to compute a prediction ...
0
votes
0answers
19 views

The mean of a class of students' weight is 120.3 pounds. The Standard Deviation is 15 [on hold]

The mean of a class of students' weight is 120.3 pounds. The Standard Deviation is 15. a.) Find the possibility of students' weight below 130 if 16 people are randomly selected. b.) construct a 95% ...
2
votes
1answer
34 views

Statistical reasoning in board game [duplicate]

In each round of the board game "The Resistance" three players are randomly and secretly chosen to be spies while the rest of the players are part of the resistance. The spies are then made aware of ...
0
votes
0answers
24 views

Find Bayes rule/action under given prior

I am able to solve for Bayes actions/rules with no data and am able to follow problems with simple data. However, I'm not sure how to solve a question where the data, $X$, is conditional on the state ...
2
votes
0answers
31 views

Expectation of a conditional density

I'm trying to figure out why the following equation holds: $$f_{Y}(y) = E(f_{Y|X}(y|X))$$ I have sort of "worked out" the RHS to be: \begin{align} f_{Y}(y) &= E(f_{Y|X}(y|X)) \\[5pt] ...
3
votes
0answers
17 views

Time series and images : difference and terminology

A time series is an ordered collection of random variables. Considering a one-dimensional time series $A_i = {a_{i1},a_{i2},\ldots,a_{it}}$ where $t$ denotes the time index. So, the time series is a ...
1
vote
0answers
30 views

How do I get initial transition matrix probabilities?

I have to develop a system to detect and prevent bank transactions fraud - just credit card transactions, for simplicity - I'm thinking about using markov chain. How would I get the initial ...
0
votes
0answers
22 views

how to normalize demand/availability matrix for Citibike data

I am not a statistician but would appreciate an outside perspective on my current project analyzing citibike data. This is a bit complicated so please bear with me. My goal is to determine to what ...
3
votes
2answers
92 views

Problem understanding the following probability problem

I have 120 blocks. Each block is one of two different materials, 3 different colors, 4 different sizes and 5 different shapes. No two blocks are exactly the same of all four properties. I take ...
0
votes
0answers
31 views

Useful references to learn the essentials of curve fitting and its application?

I know this question might be a bit too broad, but I am looking for some pointers for self-study. I am given a set of data for which I have to identify the trends, and potentially come up with some ...
0
votes
0answers
24 views

ANOVA in R Help [on hold]

I am new to this and was wondering if someone can explain to me how to get the answer for [c], [f], [g], and [h]. I understand how to get the answers for (A) , (B), (E), and (I) but theres a gap ...
0
votes
1answer
35 views

For Metropolis-Hastings algorithm, should target density and proposal distribution have the same distribution?

I watched some youtube videos about the Metropolis-Hastings algorithm. They used a Gaussian as a proposal function to estimate an unknown Gaussian, or used a Gamma function as the proposal function to ...
3
votes
2answers
35 views
+50

Next Best Product Modeling (Seeking reference material)

I am being tasked with developing a "next best product model", and at this time, I can safely say that I do not know where to start! In other words, I want to develop a model that will provide ...
0
votes
0answers
16 views

Show that canonical correlation is scale invariant

To prove this question, I defined two variables say U=a'X and V=v'Y ( X and Y are multivariate variables) having unit variances. Found their canonical correlation. I used transformations c'U and d'V. ...
1
vote
1answer
22 views

Confidence Interval difference in means Interpretation

I have a negative confidence interval ((-35.346,-8.570) for an independent samples t-test. I understand confidence intervals for the mean. However, I am uncertain about the interpretation of the ...
0
votes
1answer
18 views

Hints to find the confidence interval

Que. Consider a normal population with unknown mean $\mu$ and variance $\sigma^2=9$. To test $H_{0}:\mu=0 $against $H_{1}:\mu\ne0$, a random sample of size 100 is taken. Based on this sample, the test ...
5
votes
2answers
93 views

Combining Binomial Random Variables

(Disclaimer: This is not a homework question). I am trying to teach my self some elementary probability, and I thought of this following example: Imagine you are playing a game involving two coins. ...
2
votes
0answers
36 views

Fitting distributions to data

I have a dataset of 1000 observations of a continuous quantity, and I am trying to find a distribution that fits them best. At first I tried the Cullen and Frey graph, with bootstraping, to get an ...
0
votes
0answers
26 views

Time Series regression help

I am having trouble running my multiple regression. I can't seem to prove that the coefficients for the different variables to be statistically significant. My dependent variable is new completions / ...
0
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0answers
24 views

ARMA(2,1) autocovariance functions

I have an ARMA(1,2) model: $$x_t = 0.6 x_{t-1} + u_t + 0.1 u_{t-1} - 0.2 u_{t-2} + 5$$ and I figured out the ($\delta$ = autocovariance functions) $\delta(0)= \beta_1\delta(1) + ...
1
vote
1answer
53 views

Limiting distribution of $\frac{\sqrt{n}\left(\bar{X_n}-\mu\right)}{\sqrt{\bar{X_n}}}$ from mean of Gamma$\left(\mu,1\right)$?

Given $\bar{X_n}$ is mean of random sample with size $n$ from Gamma distribution with parameter $\alpha=\mu$ and $\beta=1$. I wanna find the limiting distribution of ...
1
vote
0answers
27 views

What I can conclude from this graph Bayesian, Euclidean and mahalanobis classifier?

I have solved this problem: Generate a set of data of N = 1000 2-dimensional vectors that stem from three equiprobable classes. The three classes are modeled by Gaussian distribution with means $ ...
1
vote
0answers
25 views

Show that weighted least squares estimator for a specific model is not consistent

Here is the background for this problem: $\qquad\qquad\qquad$ $Y_{1},...,Y_{n}$ iid $N(\mu,c^2\mu^2)$, $\,\,$ $c^2$ known. $\,$ The problem is as following: Consider the above model. Define ...
1
vote
1answer
19 views

How do I show that the mean of the posterior density minimizes this squared error loss function?

This exercise comes from Koop's Bayesian Econometrics. Given $\theta$, the parameter(s) of a model (in this case $\theta$ is a scalar), $\tilde{\theta}$, the point estimate of $\theta$, and constants ...
0
votes
2answers
51 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
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0answers
25 views

FInd the sufficient statistic

I am familiar with sufficient statistics, doing questions out of textbooks with not much problems, but ran into this question and its a very different setup, and not sure how to start going about ...
0
votes
1answer
36 views

Linear Algebra take on Chi-Squared

If I have a system where my output is the sum of my input and white gaussian noise: $Y=x+w$ and $w$~$N(0,\sigma^2I)$ now, I want to determine the distribution for $||y/\sigma||^2 = y^Ty/\sigma^2$ ...
0
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0answers
19 views

What will be the training dataset for learning a map using neural network

I am new to neural networks and training and finding it hard to understand how I can train the Neural Network (NN) in learning a time series generated by a non-linear discrete map $f : I \rightarrow I ...
0
votes
1answer
51 views

Pdf of the square of a standard normal random variable [closed]

I have this problem where I must find the pdf of $Y = X^2$. All I know is that $X$ has the distribution $N(0,1)$. What kind of distribution is $Y = X^2$? Same as $X$? How do I find the pdf?
1
vote
0answers
28 views

Hint for problem including function of random variable?

If $Y=\frac{\ln(U_1)}{\ln(U_1)+\ln(1-U_2)}$ where $U_1,U_2$ are independent $U(0,1)$ random variables, then variance of Y equals. I usually use Jenson's inequality to underestimate or overestimate ...
0
votes
0answers
18 views

Size of infinity in events

I've stumbled upon a question (with an answer) that goes as following: Let $S$ be a sample space and let $A_1,A_2,...$ be events. Define $ B_n = \bigcup_{i=n}^{\infty} A_i$. Question Show that if ...
2
votes
0answers
38 views

Markov Chain - Blackjack win problem [closed]

Got stuck with a problem from my text book. Need help please.. David is in Las Vegas, impressed by the Grandeur of the city, David decides try his luck at “Blackjack” in one of the casinos. David is ...
0
votes
1answer
17 views

Deviance Calculations for GLMs

I am having trouble calculating deviance statistics for GLMs. For example, for the exponential distribution $f(y)=\lambda*e^{\lambda*y}$ the deviance expressed in terms of responses and fitted values ...
1
vote
2answers
40 views

Identifying a confounder

I'm trying to check whether a variable is a confounder or not. Specifically, for a randomized trial where I want to investigate the effects of a reduction in class size on student performance, would ...
3
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0answers
15 views

Parametrisation invariance/covariance of the Jeffreys prior

I've been trying to understand what exactly is meant by parametrisation invariance of the Jeffreys prior. Already I've read here that invariance is technically not the best term to use, and that ...
1
vote
0answers
29 views

Are Bayes factors practically applicable?

According to Bishop's Pattern Recognition and Machine Learning on page 164, on average the Bayes factor will always favour the correct model. Given this, how can we use the Bayes factor in ...
0
votes
0answers
9 views

GLS estimator of a VAR process

I'm studiying how to derive the GLS estimator of a VAR process. I have studied the basics well, but I don't get the last passage here: Why the product can be rewritten as a quadratic form? Intuitively ...
4
votes
1answer
97 views

Marginal density and conditional density from joint density [duplicate]

I am having trouble understanding how to solve this when the variables are not discrete. Let the simultaneous density of the non-discrete stochastic variables (X,Y) be I am then supposed to find ...
2
votes
1answer
64 views

Independence of variables

Q:Two variables X and Y have same mean and variance.If U=X+Y and V=X-Y then, are U and V independent and correlated? I found that U and V are uncorrelated. But don't know how to check for ...
0
votes
0answers
14 views

Deducing the regression function using the Squared Error Loss Function [duplicate]

I am reading Elements of Statistical Learning, and came across a deduction which I cannot understand. In the second chapter, the author defines the squared error loss and deduces the conditional ...
0
votes
0answers
16 views

Skeleton of a continuous Markov Chain

I have a continuous Markov Chain with transition matrix with initial state $X_0=1$ and state space $I=1,2,3,4,5$ $$P= \begin{bmatrix} -3 & 1 & 0 & 1 & 1\\ 0 & -1 & 0 & 0 ...
3
votes
0answers
32 views

Bias-variance decomposition

In section 3.2 of Bishop's Pattern Recognition and Machine Learning, he discusses the bias-variance decomposition, stating that for a squared loss function, the expected loss can be decomposed into a ...
1
vote
1answer
41 views

Central Limit Theorem for Wilcoxon signed rank tests

Let $X_i, i=1,2,...,n$ be a set of iid observations, assumed symmetric about $\mu$. Let $R_i$ be the rank of the absolute deviations from some $\mu_0$, i.e. $R_i=\text{rank}(|X_i-\mu_0|)$. Let ...
5
votes
1answer
61 views

Compute $E\left[ \Phi \left(X \right) \Phi \left(Y \right) \right]$ for a bivariate normal distribution

Assume that $X$ and $Y$ follow the bivariate normal distribution with correlation coeffcient $\rho > 0$, zero means and scale parameters equal to one. I am looking for an elegant way to compute ...