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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20 views

Adversarial learning gradient derivation

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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0answers
37 views

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 ...
3
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1answer
61 views

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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0answers
17 views

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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1answer
40 views

Conditional probability Bivariate Normal Distribution

$\newcommand{\Cov}{\operatorname{Cov}}$$\newcommand{\Var}{\operatorname{Var}}$$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$We have that $X$ and $Y$ are random variables with a ...
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3answers
116 views

P-value word problem question

I'm having trouble understanding this question from my statistics class: I handed out a flyer about a rock show to 20 people who, at that time, did not plan on attending the show. My hypothesis is ...
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0answers
4 views

Variance of Linear Combination of Fixed Effects in Mixed ANOVA Model

Consider the mixed effects model described here. It suffices to consider only the balanced case. I need to make inference about the linear combinations of the fixed effects $L = \sum_i c_i \tau_i$ I ...
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0answers
4 views

Estimate of Fixed Effect in Mixed Effects ANOVA (Restricted versus Unrestricted)

Consider the mixed effects ANOVA model described here It is stated that the estimate of the fixed effect is $\hat{\tau}_i = \bar{y}_{i..}-\bar{y}_{...}$. But is this true for both the unrestricted ...
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0answers
20 views

Help understanding Linear Model in ESL book

Also known as "Nate slowly deciphers ESL to conceptual understanding/plainer language", part two (see part one) Help me understand this (bullets added) The term $\hat{β}_0$ is the intercept, also ...
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1answer
29 views

On finding a confidence interval. (exercise)

I am having some problems with the second point of this exercise Let $(X_1, \dots, X_n)$ be a random sample extracted from a population $X$ that is distributed as a uniform $(0, \theta)$ and let $Y_n ...
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0answers
67 views

Gradient updates for word2Vec window

I am trying to work through the first problem set of Stanfords Online course CS224, and Im having trouble with question 3(e) and getting the correct vectors for a given center word, hoping to find how ...
3
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1answer
43 views

Help understanding p-vector language

Help interpreting this excerpt (annotations (numbers in parentheses) and bullets added) from The Elements of Statistical Learning Matrices are represented by bold uppercase letters; for example, a ...
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27 views

Conditional expectation of bivariate normal

I have been reading Heckman (1979) and have tried to prove some result used (the paper points to a book which does not show the work either). I alter the notation a bit for clarity. Assume we have: $$\...
4
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1answer
52 views

Finding pdf of transformed variable for uniform distribution

This is from MITx's Intro to Probability and Statistics course, the problem is on this page. Suppose $X \sim \textrm{Uniform}(0,1)$ and $Y=X^3$. Find the pdf for $Y$. Since it's a uniform ...
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1answer
30 views

Deriving SSE of Simple Linear Regression is $\chi^{2}$

As per my notes, the key step in the proof that the sum of squares of residuals in regression is $\chi^{2}_{n-2}$ is the fact that $e_{i} = y_{i} - \hat{y}_{i}$ has a mean 0 and variance $\sigma^{2}$. ...
1
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3answers
213 views

Number of combinations for a 4- digit code lock [closed]

Your suitcase has 4-digit code lock. You forgot the code, but you do remember that the sum of first two digits was equal to the sum of two second digits. How many combinations do you need to check in ...
4
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2answers
65 views

UMVUE $g(\lambda)$ = $e^\lambda$ when $x_i \sim Pois(\lambda)$

Let $x_1 ... x_n$ be $Pois(\lambda)$ Find UMVUE of $e^\lambda$ From a previous question, I found the UMVUE of $e^{-\lambda}$ to be $(\frac{n-1}{n})^{t}$ where $t = \sum_{i=0}^n(x_i)$. $\sum_{i=0}^n(...
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1answer
14 views

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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1answer
22 views

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)? ...
4
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1answer
49 views

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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2answers
79 views

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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0answers
6 views

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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1answer
31 views

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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12 views

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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33 views

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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1answer
43 views

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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1answer
28 views

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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1answer
35 views

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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1answer
37 views

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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1answer
22 views

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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24 views

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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2answers
78 views

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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2answers
137 views

Expectation of $\frac{X_1^4}{(X_1^2 + \cdots + X_d^2)^2}$

Let $X_1$, $X_2$, $\cdots$, $X_d \sim \mathcal{N}(0, 1)$ and be independent. What is the expectation of $\frac{X_1^4}{(X_1^2 + \cdots + X_d^2)^2}$? It is easy to find $\mathbb{E}(\frac{X_1^2}{X_1^2 +...
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1answer
52 views

Finding probability assuming null hypothesis is true

Candidates 1,2 and 3 are running for a position in a company. Candidate 1 claims 38% favourability among all the voters. Assuming this is true, what is the probability that in a random sample of 500 ...
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0answers
4 views

How to convert db/s ? (smoothing factor to obtain power spectral density (PSD) of speech measured in db/s.)

Several speech features rely on an estimate of the power spectral density (PSD) of speech. To estimate the PSD, the magnitude squared DFT coefficients |X(k,ℓ)|^2 are temporally smoothed as PSD(l,k)=(...
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0answers
40 views

likelihood for binomial proportion

A man must evaluate the proportion $\pi \ (0 \leq \pi \leq 1)$ of diesel vehicles crossing a road. He notes the first 10 cars arriving and records 3 diesel vehicles. I know that $f_Y(y;\pi)\sim Bin(\...
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0answers
27 views

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 ...
2
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0answers
33 views

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 ...
3
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0answers
18 views

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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1answer
27 views

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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0answers
21 views

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 ...
2
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1answer
41 views

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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1answer
44 views

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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2answers
31 views

Probability rule involving conditional and marginal distributions

The context of my question is Baysian generative models. The text book I am reading states $p(\tilde{x} \, \vert \, D) = \int p(\tilde{x} \, \vert \, \theta) p(\theta \, \vert \, D) \, \mathrm{d}\...
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1answer
76 views

why cost function needs to be smooth, how this helps in learning?

From what I understand a smooth function 1-degree is a function whose first derivative is continuous? How this helps in estimating the parameters? What if the function is not smooth?
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47 views

Weight Decay - How to Approach this problem?

How to prove these update rules Problem In the augmented error minimization with $\tau = I $ and $\lambda > 0$ , assume that $E_{in}$ is differentiable and use gradient descent to minimize $E_{...
0
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1answer
28 views

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'...
3
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0answers
43 views

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 ...
2
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1answer
25 views

Confidence interval for Poisson distribution

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 $...
1
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1answer
27 views

Back propagation error calculation

I'm working through AI: A Modern Approach by Russel and Norvig. At the section on back-propagation, they have this to say: The idea is that hidden node j is responsible for some fraction of the ...