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15 views

Solving system of equilibrium probability state equations

I have started studying markov chains where i have these statistical equilibrium probability state equations.These equations are solved for a particular case $s_1=4,a_1=5,s_2=2, a_2=1$ and a 15*15 ...
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7 views

Definition of mutual information

The mutual information between two random variables X and Y is given by $I(X,Y) = E\bigg [ \ln\bigg ( \frac{p(x,y)}{p(x)p(y)} \bigg) \bigg]=\sum_{x,y}p(x,y)[\ln p(x,y)-\ln p(x)p(y)]$ I'm trying to ...
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0answers
23 views

Why do t statistics decrease (standard error for cofficient increases) when multicollinearity exists?

Can anybody show how the t statistic decreases when multicollinearity exists? It is easy to prove using the F test, but I don't know how to prove it using the t test.
2
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1answer
59 views

Find Variance of AR(2) process $X_i = 0.3X_{i-2} + u_i$

Full question: $X_0,X_1, …., X_n$ are distributed according to the following AR(2) process $$X_i = 0.3X_{i-2} + u_i$$ for $i=1,...,n$, $X_0=X_1=0$, and $u_i$ are iid $N(0,3^2)$. Have no idea ...
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1answer
36 views

Show that if $X$ is uniform on $[0, 1)$ then $Y = -k\log X$ is an exponential r.v. such that $⟨y⟩ = k$

Attempt: probability is conserved under change of variables. If $p(x)$ is a probability density then $p(x)dx$ is the probability that there is some $x′$ in $[x, x+dx)$. By change of variables $x$ -> ...
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0answers
14 views

How do I compute the posterior distribution of the success probability with uniform priors?

This is a question from a homework assignment A clinical trial is conducted to compare the effectiveness of three drugs. 100 patients are randomly assigned to each drug (300 total patients), and Y_1 ...
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0answers
23 views

Poisson process for queuing problem

I posted it incorrectly on a different thread so I'm reposting it here. I'm working on a problem and would love to get any advice - patients come into a clinic according to a Poisson process with ...
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1answer
47 views

Deriving the Ridge Regression $\boldsymbol{\beta}\mid \mathbf{y}$ distribution

Apparently the estimate $\hat{\boldsymbol{\beta}}$ for ridge regression comes up as the mean or mode of the posterior distribution given by $f_{\boldsymbol{\beta}\mid \mathbf{y}}$. This is the ...
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0answers
18 views

Effect and Confidence Intervals [on hold]

I am very confused about what confidence intervals means and how they relate to effects of the medians. I sent an email to my teacher asking him more about it but he sent me this back hoping that ...
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3answers
483 views

Why do we estimate mean using MLE when we already know that mean is average of the data?

I have come across a problem in textbook to estimate mean. The textbook problem is as follows: Assume that $N$ data points, $x_1$, $x_2$, . . . , $x_N$ , have been generated by a ...
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0answers
6 views

Why are the marketing-mix variables measured as period to period percent changes in the generalized bass model?

In the paper Why the Bass Model Fits without Decision Variables, Bass et al. extend the Bass model of product diffusion to incorporate marketing mix variables. For example, if the forecaster ...
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14 views

Forecast function for a MA(2) time series

For the quadratic loss function and the lead time $l = 1$, derive the forecast $z_t(l)$ for the MA(2) time series, and then calculate the prediction error $z_{t+l} − z_t(l)$ and its first and ...
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1answer
44 views

What is the probability of randomly picking 4 consecutive ladies out of a team of 11 people?

I was running the morning standup this morning and in contrary to the usual alphabetical order I announced I was going to go in "random" order today, however I then ended up giving all the ladies of ...
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0answers
20 views

Conditional distribution

Given the two discrete random variables $X_1$ and $X_2$, I want to find $P(X_1=x_1,X_2=x_2|X_1+X_2=p)$. As a first step, is it correct to argue that: ...
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11 views

Paired or Unpaired

As a part of curriculum we had a research project to find out that turmeric is effective on oral wound. We took two groups where experimental group received turmeric only for seven days and control ...
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0answers
21 views

Geometric proof of partical correlation with pythagoras theorem [on hold]

Please help me with my statistics lesson. I need to prove the partical correlation formula with the definition and the pythagoras theorem. The definition (partical correlation): ...
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3answers
148 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 ...
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1answer
30 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 ...
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0answers
10 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: ...
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2answers
48 views

Probability that 2 out of 3 friends win a raffle?

Let's say it's me and 2 other friends are at a raffle. There are 10 total participants. 3 of the 10 of us will win something, I don't care about what we win. A person can't win more than once. What is ...
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1answer
54 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 ...
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1answer
47 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, ...
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8 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 ...
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19 views

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

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% ...
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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 ...
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0answers
29 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 ...
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37 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] ...
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0answers
22 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 ...
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0answers
31 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 ...
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24 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 ...
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2answers
95 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 ...
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32 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 ...
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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 ...
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3answers
64 views

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 ...
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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. ...
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1answer
23 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 ...
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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 ...
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2answers
97 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. ...
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0answers
38 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 ...
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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 / ...
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0answers
25 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) + ...
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1answer
54 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 ...
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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 $ ...
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0answers
27 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 ...
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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 ...
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2answers
55 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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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 ...
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1answer
37 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 $||y/\sigma||^2 = y^Ty/\sigma^2$ ...
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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 ...
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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?