# Tagged Questions

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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### 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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### 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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### 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.
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $||y/\sigma||^2 = y^Ty/\sigma^2$ ...