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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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1answer
28 views

How to check covariate is balanced?

I try to check if the covariate is balanced by computing the absolute pooled standardized difference, but I don't know how I could get the sample variance for my treated and control covariate? any ...
5
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1answer
184 views

Testing correlation and the t-statistic used in Simple Linear Regression

Given $H_0$ : $\rho=0$ and $H_A$ : $\rho\neq0$, we use the test statistic $t_{n-2}$ , which is $\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$. I have to show that $\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$ equals $\frac{\...
3
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1answer
39 views

PDF of the minimum of a geometric random variable and a constant

I have $X \sim Geo(p)$, such that $$p(x) = (1-p)^{k-1}p, \ \ x = 1,2,3, \ldots$$ and Y is a constant random variable which assumes the value of the constant integer $t$, such that $$P(Y=t) = 1, \ \ ...
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0answers
14 views

slope coefficient- Sampling distributions [on hold]

I have a question in my exam, which I do not know exactly the answer, Can you please guide me? Q: Do you believe the sampling distributions for the slope coefficients are at least approximately ...
2
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1answer
214 views

Bayes optimal decision for logistic regression: Self-study exercise

We want to find the Bayes optimal decision for logistic regression. That means that the goal is to find the actions, which minimize our expected loss (also often called expected cost or risk). Here ...
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0answers
26 views

UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution

I have a random variable X with pmf $$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$ and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$. Here is my attempt so ...
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0answers
52 views

Calculation of a time until the absorbing state given chemicals are initially separate

Biochemist from Siberia has just patented a new super genome formed by amalgamating genome X and genome Y. These genomes are harmless when separate, but when mixed there is a 45% chance of an ...
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18 views

Normal Distributions Definition [on hold]

Confirm the bivariate Gaussian mean and covariance by explicitly evalu- ating the exponential integral using the definition of the multivariate Gaussian density function
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1answer
304 views
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1answer
17 views

Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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0answers
31 views

Random sampling

Problem: We draw a random sample of size n from a population of N. The i-th individual of the population will be chosen with probability: $$\frac{n-n_{i}}{N-i+1}$$ Where $n_{i}$ is the number of ...
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1answer
26 views

problem regarding time series modeling using R

I have this time series data and my aim is to fit a time series model. When i plot the time series data , it seems to be data is not stationary. These are the plots based on the first difference, ...
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0answers
14 views

Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3) [duplicate]

Let us consider the random vector $X=[X_1,X_2,X_3]$, which follows a multivariate normal distribution. That is, for each entry $X_i$: $X_i \sim N(\mu_i, \Sigma_i)$. What I am trying to compute is the ...
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1answer
194 views

What is the benefit of using asymmetric LDA prior?

I am reading the paper Rethinking LDA: Why Priors Matter. The author claims that the combination of asymmetric prior for document-topic proportion and symmetric prior for topic-word is the best, ...
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1answer
37 views

Relation between binomial and negative binomial

I was reading on negative binomial from a Statistics textbook and came across this portion on probability relation between binomial and negative binomial. $Y$ refers to the number of trials required ...
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1answer
14 views

Getting 2 defectives

An item is produced by a machine in large numbers. The machine is known to produce 5% defectives. A quality control engineer is testing the items randomly. What is the probability that at least 5 ...
0
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1answer
17 views

2-sample bootstrap hypothesis test - comparing locations but different estimators in two samples

I have two independent samples X and Y where $x_i \sim F$ and $y_i \sim G$. Two different estimators A and B map X to $x_0$ and Y to $y_0$ respectively. I'd like to compare $x_0$ and $y_0$. The ...
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1answer
30 views

Finding the probability for non defective battery

A car manufacturer purchases car batteries from two different suppliers A and B. Suppose supplier A provides 60% of the batteries and supplier B provides the rest. If 6% of all batteries from supplier ...
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1answer
23 views

What is the probability that two samples from a source have no overlap?

I picked 29 results from a list of 429 results. I then picked a second group of 27 (with replacement) results from the same list of 429. There was no overlap between the two samples. What is the ...
3
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1answer
49 views

Properties of Kernel Density Estimators

Given Let $X \in \mathbb{R}$ be a real-valued random variable with theoretical probability density function (pdf) $f(x)$ and corresponding cumulative distribution function (cdf) $F(x)$. Let $X_1, X_2,...
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0answers
23 views

Expectation of kth order statistic of Pareto distribution

I am trying to find the expected value of $X_{(k)}$ Given cdf $$ F(x) = \begin{cases} 1-\left(\frac{\sigma}{x}\right)^\alpha, & x > \sigma\\ 0, & \text{else.} \end{cases}$$ My attempt: $$...
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0answers
31 views

Finding UMVUE for $g(\theta)$ that satisfy $g(0)=0$ in discrete uniform $f(x\mid\theta)=\frac{1}{\theta} I_{1,…,\theta}(x)$

Let $x_1, \ldots x_n, \overset{\text{i.i.d}}{\sim}f(x\mid \theta)=\frac{1}{\theta} I_{1,...,\theta}(x)$. I know that $T=X_{(n)}$ is complete and sufficient statistic for $\theta$ and $$f_T(t\mid\...
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1answer
40 views

order of the time series model

I am kind of new to time series modeling. I am trying to fit a model for a time series variable. I am trying to fit a ARMA model. I am using R to do the analysis. When i estimate the model using both ...
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0answers
35 views

Showing that $(l'\Sigma l)\chi_p^2 = (l'S l)\frac{(n-1)p}{n-p} F_{p,n-p}$

Let $X_1,\cdots X_n$ be i.i.d. $N_p(\mu,\Sigma)$. I have that when $\mu$ and $\Sigma$ are unknown that the Scheffe type method gives $(1-\alpha)100$% confidence intervals for all linear combinations $...
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1answer
624 views

Why is the variance of the error term (a.k.a., the “irreducible error”) always 1 in examples of the bias-variance tradeoff?

I'm reading Introduction to Statistical Learning. The relevant part is referenced here: Proof/Derivation of Residual Sum of Squares (Based on Introduction to Statistical Learning). When the author ...
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1answer
198 views

Is the following model parameterization identifiable?

Let $X_i's$ be independent $i=1,2...n$ with $X_i\sim N(\mu+\alpha_i, \sigma^2)$ for each $i$. Let $\theta=(\alpha_1,...\alpha_p,\mu,\sigma)$ and $P_\theta$ be the joint pdf of the $X_i's$. So, $P_\...
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0answers
4 views

Plotting Custom PDF in R [migrated]

Given: Consider the density function $\phi$ defined on $\mathbb{R}$ for $a\in \mathbb{R}$ and $b \in \mathbb{R}_+^\star$ such that $\forall x \in \mathbb{R}$, $$ \phi(x; a,b) = \frac{1}{\sqrt{2\pi b^...
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3answers
189 views

Learning material for regression analysis

I found a lecture notes from regression analysis but it was quite hard to learn from it. Those notes were aimed for students who had read just basics of statistics beforehand. I have a background in ...
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0answers
47 views

Intuition about a coupon problem were we ask for the distribution of the unique coupons when the number of draws is fixed

Alternative viewpoint of the coupon collectors problem In the coupon collectors problem we draw from a collection of $n$ coupons, with replacement and ask the question how many draws $K$ it takes to ...
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2answers
106 views

Questions on likelihood analysis

Whilst studying likelihood methodologies, I've come across some results that I haven't been able to work out. If $X$ and $Y$ are Poisson with means $\mu_{X}$ and $\mu_{Y}$, then the conditional ...
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0answers
21 views

Derivation of AMISE and Bandwidth

Given: Let $K(\cdot)$ be a bona fide kernel. Let $f$ be a pdf and $\widehat{f}_n$ is kernel density estimator with bandwidth $h$ based on a sample $X_1,X_2,\cdots,X_n$ of size $n$ draw iid from $f$. ...
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1answer
30 views

Why do we use “Sum of Squared Errors” as loss function in linear regression? [duplicate]

What is a loss function? How can we relate the slope of Linear Regression with Sum of Squared Errors?
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1answer
77 views

Finding $\mathrm{Var}(N)$ if $N=\inf\{n\ge1:\sum_{i=1}^nX_i>1\}$ where $X_i$'s are i.i.d Exponential variables

Suppose $X_1, X_2, X_3, \ldots, X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$. If $S_n = X_1 + X_2 + \ldots + X_n$ ...
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1answer
9 views

Markov Chain Question/Notation Confusion

Show that if $(X_n)_{n \geq 0}$ is a discrete-time Markov chain with transition matrix $P$ and $Y_n = X_{kn}$, then $(Y_n)_{n \geq 0}$ is a Markov chain with transition matrix $P^k$. I am a little ...
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0answers
18 views

Robust Trimmed Mean Proof

This question arises from the first chapter of "ROBUST LEARNING: INFORMATION THEORY AND ALGORITHMS" by Jacob Steinhardt. Given is the following: there are $\epsilon n$ outliers and $(1-\epsilon)n$ ...
1
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1answer
50 views

$E[\bar{X^3}]$ of N(μ,1)

Suppose $x_1, x_2, x_3,\ldots, x_n$ i.i.d. Normal(μ , 1) random variables with μ $\in$ $\mathbb{R}$ how can we calculate $E[\bar{X^3}]$
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56 views

auto-correlation and OLS regression

I was trying to find the OLS estimator for the model: $Y$ = $\beta_0$ + $\beta_1X_{1t}$ + $\beta_2X_{2t}$ +.......+ $\beta_5X_{5t}$ + $e$ t = 1,2,3 ......, 50 time ordered observations X is a full ...
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2answers
34 views

Hazard function equality $T = \min(T_1, T_2, …, T_n)$ [on hold]

Let $T_i, i = 1, ..., n$ be independent continuous random variables. Denote by $h_i(t)$ the corresponding hazard function of $T_i$. Let $T = \min(T_1, T_2, ..., T_n)$. Denote by $h_T(t)$ the ...
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0answers
68 views

Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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2answers
44 views

Checking if a minimal sufficient statistic is complete

Let $X_1, \cdots, X_n$ be iid from a uniform distribution $U[-\theta, 2\theta]$ with $\theta \in \mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete. I found ...
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1answer
1k views

Deriving K-means algorithm as a limit of Expectation Maximization for Gaussian Mixtures

Christopher Bishop defines the expected value of the complete-data log likelihood function (i.e. assuming that we are given both the observable data X as well as the latent data Z) as follows: $$ \...
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2answers
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Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
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1answer
210 views

Hidden Markov model - formulas

I am self-studying Kevin Murphy's book, and trying to understand the math behind the HMM. I am struggling with the following derivation. Why can we in 17.46 cross out the $X_{1:t-1}$ term? my guess ...
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1answer
278 views

When is further reduction than the order statistics not possible?

Let $X_1, ..., X_n$ be a random sample from a population with location pdf $f(x-\theta)$. Show that the order statistics $T(X_1, ..., X_n) = (X_{(1)}, ..., X_{(n)})$ are a sufficient statistic for $\...
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0answers
6 views

Show that slope for segmented regression changes at threshold z

I am studying a variation of regression model called segmented regression. For a response variable $y$ and covariate $x$, it is set that the first $m$ values of $x$ are less than threshold $z$ and the ...
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0answers
5 views

How to choose p in the RESET test?

The RESET test allows us to detect whether the linear functional form assumption is correct or not by using the auxiliary regression: \begin{align} y_t = \alpha_1 + \alpha_2 \hat{y}_t^2 + \alpha_3 \...
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0answers
28 views

Accuracy and ROC for Logistic and Decision Tree

So I run a logistic regression and decision tree model using same data. The accuracy shows that the decision tree outperforms logistic slightly. However, my ROC curve shows that logistic is much ...
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0answers
18 views

Heteroskedasticity-robust White estimator

The heteroskedasticity-robust White estimator is defined as: \begin{align} V_{\hat{\beta}} = (X'X)^{-1}\left(\sum_{i=1}^n x_i x_i' \hat{e}_i^2 \right)(X'X)^{-1} \end{align} with X the matrix of ...
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0answers
24 views

How do we test a hypothesis of a specific number from a regression?

I have a homework assignment that has regression results that may or may not make sense. There are 3 variables: price, bathrooms, and bedrooms. Doing a univariate regression of bathrooms and ...
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2answers
25 views

confidence interval of function of parameter

Self Study Question: We already know that the confidence interval of a parameter M is (-1,2). We are supposed to find the confidence interval of 1/M. The function is 1/X, which is One to One, so ...