# 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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I'm working through Convolutional Neural Network paper here on adversarial learning and I'm having trouble with the derivative proof of adversarial logistic regression. The correct answer presented (...
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### Jointly Complete Sufficient Statistics: Uniform(a, b)

Let $\mathbf{X}=x_1, x_2, \dots x_n$ be a random sample from the uniform distribution on $(a,b)$, where $a < b$. Let $Y_1$ and $Y_n$ be the largest and smallest order statistics. Show that the ...
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### Deriving the normalizing constant for the multivariate Gaussian

I am trying to derive the normalizing constant for the multivariate Gaussian. The book I'm following suggests diagonalizing the covariance matrix and then using a change of variables. So, we consider ...
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### multivariate normal distribution. And marginals distributions [duplicate]

if $X_1$ and $X_2$ are random variables that marginally have a normal distribution, is it true that the joint $(X_1,X_2)$ has a distribution according to a multivariate normal density distribution? ...
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### Independent in statistics and independent in linear algebra?

In statistics it says two random variables are independent if and only if $F_{X,Y}=F_X(x)F_Y(y)$ If linear algebra it says: Two or more functions (random variable is a function), equations, or ...
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### Probability of drawn cards coming in an increasing sequence

What is the probability that 13 randomly drawn cards will come in a strictly increasing sequence of ranks in a common 52 card deck (239KA is strictly increasing, but A239K is not, and JJJKK is not)? ...
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### Constrained optimization algorithm in linear regression

I am interested in the following constrained parameter estimation in linear regression, $$\min_\beta\sum_{i=1}^{n}(y_i-x_i\beta)^2 + \lambda \sum_{j}^{p}f(\beta_j)$$ where the model is $y=x\beta+e$, ...
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### Truthfulness of statements on the expected values of random variables

Are these statements true or false? Why? $E(|X|)\le 1 + E(X^2)$ $0≤|x|<1+x^2$ for all choices of $x$ with $x$ real number. What with $X$ random variable? if $E(X)<0$ and $\theta \neq0$ ...
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### How to formulate moving average model in the problem of estimation

The source input $\mathbf{x}$ is in the form of the feature vector having $d$ components of length $d$ is transmitted via a channel whose impulse response is modeled as moving average (finite response,...
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### Event of pulling a white ball in first draw is not independent of pulling a 2nd white ball

You have two urns. Urn One contains 6 white balls and 2 black balls. Urn Two contains 7 white balls and 8 black balls. You pick a random urn; you get Urn One with probability 2/3. You pull out, ...
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### counting process and birth-death process

This is from notes: I have some questions What does it mean by $X(t_2)>X(t_1)$? Let's say there are state i and j and $X(t_2)=i$, $X(t_1)=j$, what does it mean by $i>j$? 2.For the birth-...
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### Probability that 7 random cards will contain at least 4 spades

Consider a common 52 card deck. What is the probability that 7 random cards will contain at least 4 spades? Solution There’s $52\choose7$ ways of picking 7 cards from the deck; there’s $13\choose{i}$ ...
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### Neural network basic concepts

I want to apply neural network as an auto associative memory. So, the desired output is equal to the input. I would apply Hebbs rule to train the network. I have a pattern in the form ...
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### Probability of more than 5 responses surveying two locations with different response rates

You are looking for people between 16 to 23 years to answer a survey. Assume houses on First Street, based on copious amounts of empty beer bottles in front yards and your past experience, has a ...
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### How to show association between categorical variables with graphs?

I have 4 categorical variables in a dataset let $X_1$, $X_2$, $X_3$ and $Y$. Where $X_1$,$X_2$ and $Y$ is a binary random variable and $X_3$ have 3 categories. How can I study the association of $X_1$...
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### Poisson Regression model in example

I need to fit a model to study the severity of bicycle accidents. The data description is $X_1$: is the use of helmet, a categorical variable (1=Yes,0=No) $X_2$:the speed at which he was riding, a ...
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### MLE of exponential distribution

Let $Y\sim Exp(1)$ and $T=\mu+Y,\ \mu\in \mathbb{R}$. Let $t_1,\dots,t_n$ be a simple random sample from $T$ with $\mu$ unknown parameter. How can I find MLE for $\mu$? I know that the likelihood ...
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### Find MLE on custom density function

Let $y_1,\dots,y_n$ be i.i.d. random variables from $$p_{Y_i}(y_i;\alpha,\beta)=exp\{y_i(\alpha+\beta x_i)-ln(1-e^{\alpha+\beta y_i})\},\ y_i=0,1$$ $\alpha, \beta$ are unknown real parameters ...
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### Expected value of MLE of uniform distribution [closed]

Let $X_1,\dots,X_n$ be a simple random sample from $U(0,\theta)$. Let $\hat\theta=X_{(n)}$ be the MLE estimator. How can I find the expected value of $\hat \theta$ and prove that is it consistency? ...
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### Parametric bootstrap testing for random effect in GLMM

I am trying to learn GLMM using R. From what I have understood parametric bootstrap is a robust method to know the significance of fixed as well as random effects ...
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### Intuition behind a conditional probability problem [duplicate]

I was watching the Harvard's statistics course and came across this problem (https://m.youtube.com/watch?v=JzDvVgNDxo8) 9:30. Pick two cards from a deck of 52 cards what is the probability of getting ...
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### Relating sufficient statistics to parameters

I'm studying sufficient statistics and I came across this problem: A dataset consists of independent triples $(W_i,Y_i,Z_i)$ of independent random variables with distributions as follows,  W_i \...
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### Bayesian Probability - Hybrid approach to calculate components?

The task The following question is from a Basic Statistics course on Coursera: In a shop, people can take chewing gum from a dispenser on the right, or the left. The dispenser on the right has 7 ...
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### Feature map of kernel

Let $K_1,K_2$ be valid kernels. Kernel $K_1$ has a feature map $Θ(x)∈R^{50}$ while Kernel $K_2$ corresponds to feature map $ψ(x)∈R^{10}$, satisfying: $∀i=1..10,∀x:ψ_i (x)=0.2 Θ_i (x)$ What is the ...
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### Logistic regression variable selection methods

I'm having trouble to understand Backward elimination in Logistic Regression model. I was looking at this example of Agresti, Categorical Data Analysis, to see how Backward elimination works. What ...
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### Probability that the p-value is less than 0.05 if H0 is true?

Suppose that under $H_0$, a measurement $X$ is $N(0,\sigma^2 )$, and that under $H_1$, $X$ is $N(1,\sigma^2 )$ and that the prior probability $P(H_0) = P(H_1)$. With $\sigma$ = 1, what is the ...
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### standard deviation in calculating confidence intervals

I asked a question earlier and realized the question I asked wasn't a good representation of what I meant to ask. (Why use the standard deviation of sample means for a specific sample?) Now that I'...
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### Error measure and learning process - Stuck!

Problem : You have N data points y1 <= ... <= YN and wish to estimate a ' representative' value. 1) If your algorithm is to find the hypothesis h that minimizes the in sample sum of squared ...
Let $y_1, y_2, \ldots,y_n$ be a simple random sample from a random variable $Y \sim Po(\lambda)$. I should calculate: $\mathbb{E}(S_n)$ and $\mathbb{Var}(S_n)$, where $S_n=\sum_{i=1}^{n}Y_i$ and \$...