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Questions tagged [indicator-function]

A function defined on a set indicating membership in a subset and taking values 0 and 1.

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Expectation & Covariance matrix of indicator vector

Suppose we have the $p$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$. Take the set $A$ to be (without loss of generality) the negative real line, thus $A = (- \...
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Comparing truncated distributions based on mean and cdf

Let $\tilde{x}$ and $\tilde{y}$ be random variables with pdfs $f_x(x)$ and $f_y(y)$ and cdfs $F_x(x)$ and $F_y(y)$. Given that $E[\tilde{x}] \geq E[\tilde{y}]$ $F_y(c) \geq F_x(c)$ for all $c \in \...
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Dealing with 0's in loglog regression by using indicator functions I(x > 0)?

Assume we want to estimate the following model $y = e^{\beta_0} * x_1^{\beta_1} * x_2{\beta_3}$ which we can linearize into $\log(y) = \beta_0 + \beta_1 * \log x_1 + \beta_2 * \log x_2$ Assume that ...
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Indicator variables in a Linear regression

For linear regression analysis, will changing the indicator variables change the table of sums of squares? Is there a way to illustrate this through a mathematical way via statistical formula or ...
confusedaboutstats's user avatar
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Distribution of the Sum of Dependent Bernoulli Random Variables

Suppose that we have two random variables consisting of an indicator function like below: \begin{align*} V_1=1[c_1\geq u_1] \\ V_2=1[c_2\geq u_2] \end{align*} where $c_1$ and $c_2$ are constants, and $...
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Expectation of number of different "blocks" [duplicate]

Suppose we have 52 decks with 26 red and 26 black cards. We shuffle them at a random order. Then we define a "block" as cards with same colors, for example, BRRB has 3 blocks and BRRRBBRRRR ...
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Use indicator function in regression equation

I have 2 regression equations: 1st Regression: $I_t = a_1 + \beta_1E_t+X'_t\delta_1+\epsilon_1$ 2nd Regression: $I_t = a_2 + \beta_2E_t+X'_t\delta_2+Y'_t\lambda_2+\epsilon_2$ The two regressions are ...
miamialan's user avatar
3 votes
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Compute likelihood associated with indicator function and MLE $\lambda$

Let $Y_1, . . . , Y_n$ be a random sample from a Poisson distribution with parameter $\lambda > 0$. One observes only $W_i = I_{Y_i>0}$ (indicator function). Compute the likelihood associated ...
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From density function to cumulative distribution function?

Given $$f(y)=\theta/(\delta^{\theta}y^{\theta+1})\mathbb{1(y>1/\delta)}$$ where the last factor is the indicator function, and I am asked to compute the Cumulative Distribution Function of y: $$\...
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Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold?

Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold? (Here $\mathbf{1}(\cdot)$ is the indicator function). To me it seems that it holds. Here is the proof: $E(X|X\in A)=\int_{-\infty}^{\...
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What is a differentiable approximation to the indicator function a != b [closed]

I am dealing with an optimization problem where I'd like to regularize two parameters $a$ and $b$. The penalty should be $1$ if the parameters differ and $0$ if they are the same. The motivation is ...
idomorphism's user avatar
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Conditional expectation conditioned on an Indicator variable

Suppose I have a random variable $u$ that is standard uniformly distributed. And I have an indicator variable $S_{i}=1\left(V_{i}>0.5\right)$. Now I am interested in the following conditional ...
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Expected Value and Indicator Random Variable

In this ques P(1st person selects his own hat) = 1/N, but then the next person selects his own hat, he is picking from the pool of 'N - 1' hats. Why do we treat P{Xi = 1} = 1\N for all i? Shouldn't it ...
stats_geek's user avatar
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Expectation of inverse of sum of i.i.d. positive variables [duplicate]

Description: There is an indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time ...
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Expectation of inverse of sum of iid random variables [closed]

Description: There is a indicator function called $I_{i}^{k}$ as follows: \begin{align*} \begin{split} I_{i}^{k}= \left\{ \begin{array}{lr} 1, \; {\rm if \; the \; event \; happened\; at\; time \...
wenting liu's user avatar
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1 answer
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Indicator function with equal sign for probability integral

In the beginning of the book Train (2009, p.4) on "Discrete choice methods with simulation" we read: Define an indicator function $I[h(x,ε) = y]$ that takes the value of $1$ when the ...
garej's user avatar
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Integrating with considering two indicator function

Consider exponential random variables $X$, $Y$, and $Z$ with $\lambda_x$, $\lambda_y$, and $\lambda_z$, respectively. Now I want to calculate the following integration: $$E[X1_{\{X<Y\}}1_{\{X<Z\}...
Optimized Life's user avatar
3 votes
2 answers
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Finding a critical region for a mixture

Let $\ X_1 , X_2 $ be two iid random variables with normal N ( $\theta,1 $) distribution. Further , consider bernoulli random variable V with P(V=1) = $ \frac{1}{4} $ and which is independent of $\ ...
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Independence of variables in expectation

I know that $X$ and $Y$ are independent, and have an expression $$E[I(Y>X)*I(X>2)].$$ Is the independence between $X$ and $Y$ enough to say that $$E[I(Y>X)*I(X>2)] = E[I(Y>X)]*E[I(X>...
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How to prove or disprove that a complete sufficient statistic exists?

We have a discrete random variable which takes values with probabilities $p, q, p+q$ and $r$. I want to construct a complete sufficient statistic based on a single observation from this distribution, ...
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How can I solve this exercise? (Variance with indicator function)

I am trying to compute the variance of T but I am having some troubles. T is defined like $T=\sum_{i=1}^m Y_i$ and $Y_i=YI(U_i \leq \varrho) + Y_i^oI(U_i >\varrho)$ where I(.) is an indicator ...
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Regression methods for indicator function as covariate

I am looking for a regression method to fit the following model: $$Y=\beta_0 + \beta_1X+\beta_2 I(X>\beta_3)X + \varepsilon,$$ where $\varepsilon \sim N(0, \sigma^2)$, and $I$ is the indicator ...
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Prove $P(A\cap B)>P(A)P(B)$ if $A$ and $B$ are independent conditional on $C$ and $P(A|C)>P(A|C')$ and $P(B|C)>P(B|C')$

I want to prove that $P(A\cap B)>P(A)P(B)$ given: $P(A\cap B|C)=P(A|C)P(B|C)$ $P(A\cap B|C')=P(A|C')P(B|C')$ $P(A|C)>P(A|C')$ $P(B|C)>P(B|C')$ $P(C)\neq 0,1$ Where $C'$ is the compliment of ...
Professor Reichenbach's user avatar
5 votes
2 answers
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When are these functions of a random variable independent?

Assume that $I$ is an indicator variable \begin{equation} I=\begin{cases} 1 &, \text{if} \,X<0 \\ 0 &, \text{else}\end{cases} \end{equation} and $X$ is a random variable. I want to know if $...
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Help regarding the notation used in certain expression for the pdf

The probability density function of a certain random variable is given by $$ \begin{aligned}g(t)&=4 n(n-1)\\&\int_0^1 x(x+t)1(x+t<1)[(x+t)^2\,1(x+t<1)+1(x+t\ge1)-x^2]^{n-2}\, dx\,1(t>...
AgnostMystic's user avatar
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3 answers
2k views

indicator function in objective function with $L_2$ norm

I am trying to solve an optimization problem. The objective function is as follows: $\arg \min \lVert\mathbb{A}\mathbf{x} - \mathbf{b}\rVert^2 + \text{other linear least squares terms} + \mathcal{I}(\...
Jogging Song's user avatar
2 votes
1 answer
252 views

What does it mean to set a dummy variable to unity?

I am reading a paper where a regression model is introduced. The specific model is irrelevant but it has a dummy explanatory variable, about which the authors say the following: I is a dummy variable ...
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Distribution of sum of Indicator Variables

Let $X1 $, $X2 $, $X3 $...$Xn $ be n observations with distribution function $F $. Let $F^{*} $ be the empirical distribution of the random sample. $F^{*} = \frac{1}{n} \sum I(X_{i} \le x)$ where I = ...
Dom Jo's user avatar
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1 answer
670 views

What does it mean to represent a word as an "indicator vector" in natural language processing?

I'm reading the paper A Neural Attention Model for Sentence Summarization (Rush et al., EMNLP 2015) and came across the following: Let the input consist of a sequence of $M$ words $\mathbf{x}_1, \...
Sean's user avatar
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1 vote
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Markov models and occupation time

I'm presented with the following explanation and proof: Let $(X_n)$ be a Markov chain, and fix a state $j \in S$. Define indicator variables: For $n = 0, 1, \dots$, let $$I_n(j) = \begin{cases} 1 &...
The Pointer's user avatar
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5 votes
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What is the variance of the difference of two random-variable indicators with a chance of intersection between events?

Let $X$ be an event whose probability P($X$) = $p$ and let $Y$ be an event whose probability is P($Y$) = $q$. The probabilit$Y$ of $X$ intersection with $Y$, $P(X \cap Y)$ = $r$. $I_X$ is the ...
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Convolution of two indicator functions [duplicate]

I am experiencing difficulties understanding the concept of convolution and especially convolution of 2 independent indicator functions. I would be really thankful if someone can help understand the ...
debell's user avatar
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Conditional linear regression with indicator variables (Python)

I have the sample dataset below 20 observations of Y variables and 20 observations of X variables. Both are normalized (z-scored). I have a prior that (i) larger magnitude X values with $abs(X) >= ...
qwer's user avatar
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1 answer
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Calculate a Conditional Expectation via Samples

Consider a binary random variable $Y \sim p(Y)$, a random variable $X \sim p(X)$ (can be discrete or continuous) and a conditional distribution $p(X|Y)$. Suppose that I generate $N$ samples from $p(Y)$...
KRL's user avatar
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1 answer
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An trick involving indicator variables, conditional probabilities and expepectation

I am reading a paper on a statistical model for credit risk management. The details of the model are not important, but I mention for context. Suppose that we are interested in a stochastic process ...
TheRefrigerator's user avatar
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1 answer
161 views

Indicator function expression for two sample test

I am trying to understand the theory of this paper. Basically, the paper tries to lay down a framework for using two sample tests using binary classifiers. Let there be two samples $S_p$~$P^n$ and $...
bandit_king28's user avatar
1 vote
1 answer
229 views

Expected value using indicator variable

Suppose that $8$ white balls and $2$ black balls will be randomly ordered, from left to right (with all permutations of the $10$ balls equally likely), what is the expected value of the number of ...
rgk's user avatar
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1 vote
1 answer
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Mean and variance of probability density with multidimensional indicator function

I encountered the following question while studying machine learning: We are asked to calculate mean and covariance of a given probability density function $$p(x) = \frac{1}{16} \cdot 1_{0 \leq x_1 ...
BlockchainDieter's user avatar
1 vote
0 answers
358 views

How to solve an optimization problem with variable in indicator function?

How to solve the following optimization problem? $$ \underset{D, U \in \mathbb{R}}{\min} (1+a)\text{E}_{X}[(X-D)\cdot\mathbb{1}_{\{D<X\leq U\}}] - (M-D)\cdot\mathbb{1}_{\{D<M\leq U\}}, $$ where ...
Xinyuan Wei's user avatar
2 votes
1 answer
2k views

Indiscpecies - multipatt and overcoming multi-comparrisons

What is the best way to overcome the potential issue of type 1 error when doing an indicator analysis using the multipatt function in the ...
Elle Bowd's user avatar
1 vote
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2k views

How do I get the CDF given a PDF with an indicator function?

Given $$f(x)=\frac{3}{4}(1-x^2 )𝟙_{(-1,1)}(x),\,x\in\mathbb{R}$$ How do I get $F(x)$? I know that the CDF of a random variable is defined as $$F(a)=\int_{- \infty}^{a} f(x)\,dx$$ In my case this ...
Zweifler's user avatar
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2 votes
1 answer
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How does one treat the indicator function algebraically?

I am supposed to find the value for which $f(\cdot)$ (I don't understand the use of the dot here either) is a PDF in the following function: $$ f(x) = k(1-x^2)𝟙_{(-1,1)}(x), x \in \ R $$ I know I ...
Zweifler's user avatar
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Formulating the Netwon Raphson

If this is the dataset under consideration and ...
Emily Fassbender's user avatar
7 votes
2 answers
2k views

What does 1 with an inequality in the subscript mean? [duplicate]

I have not seen the following notation before (the 1 with a subscript in the density): Consider the problem of sampling from the truncated normal distribution $\mathcal{N}_t (\mu, 1, a)$, given ...
jm22b's user avatar
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Central Limit Theorem Approximation Exercise

Every time Sherry buys a book it is a math book with probability 0.3 and a novel with probability 0.7. Sherry keeps her math books in a bookcase that can hold 50 math books and keeps her novels in ...
queence's user avatar
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6 votes
1 answer
4k views

Fisher's Information for Laplace distribution

Say we have $f(x , \theta) = \frac{1}{2}e^{-|x-\theta|}$ Lets assume for simplicity, we only have 1 sample. We find that the log-likelihood for this distribution is: $$ l(\theta , x) = -log(2) + (\...
rannoudanames's user avatar
1 vote
1 answer
2k views

concept of binary indicator function [closed]

In my textbook it talks about the binary indicator function. It says "The distribution represents the fact that X is always equal to the value 1, in other words, it is a constant". Is big X the state ...
asilvester635's user avatar
1 vote
0 answers
90 views

Densitiy of order statistic when in a certain interval

$X$ is distributed with $F$, i.i.d. and with densities. I am trying to discern an expected value for a certain order statistic $X_{k}$ under the condition that $X_{k}$ is closest to some value $\...
IMA's user avatar
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7 votes
1 answer
566 views

How to compute expected values of compound events?

A helpful hint would be appreciated because I cannot seem to figure out how to calculate the expected value A lot contains 17 items, each of which is subject to inspection by two quality assurance ...
bab354's user avatar
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5 votes
1 answer
790 views

Undefined term in log likelihood

I am trying to compute the MLE for a sample $X_1, \ldots, X_n$ where $$ f(x,\theta) = (\theta + 1)\theta^x, 0 \leq x \leq 1 $$ I have defined the likelihood as: $$ \begin{align} L(\theta|x) &= \...
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