Questions tagged [independence]

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

Filter by
Sorted by
Tagged with
1
vote
0answers
19 views

Are two uncorrelated Rademacher rv always independent?

Let $X$ and $Y$ be uncorrelated Rademacher random variables Then, $Cov(X,Y)=E[XY]-E[X]E[Y]=0$ But $E[X]=E[Y]=0$ Then $E[XY]=P(X=1,Y=1)+P(X=-1,Y=-1)-P(X=1,Y=-1)-P(X=-1,Y=1)=0$ I know that $P(X=1)...
1
vote
1answer
59 views

show that A and B are independent

Given $P(A,B,C,D) = P(A)P(B)P(C|A,B)P(D|C)$ show that $P(A,B) = P(A)P(B)$ First one can write $P(A,B,C,D) = P(A,B)P(C,D|A,B) $ hence $P(A,B)P(C,D|A,B) = P(A)P(B)P(C|A,B)P(D|C)$ $P(A,B) = \frac{P(A)...
3
votes
3answers
78 views

What does the minimum of a random variable mean?

Let $X_1, X_2, X_3, \cdots,X_n$ be independent and identically distributred (iid) random variables. Then, how would you know/calculate what $min(X_1, X_2, X_3, \cdots,X_n)$ is?
4
votes
1answer
36 views

What does it mean when we say the estimators need to be independent when using Ensemble Methods in Machine Learning?

In collective learning (ensemble methods) we need the estimators to be Independent/ uncorrelated from one another. Do I understand correctly, that this means we need to draw the data samples without ...
0
votes
0answers
6 views

How to force hidden code in autoencoder to produce independent code units?

I am trying to construct an undercomplete autoencoder with number of hidden neurons in middle layer (bottleneck) much lower than number of input (and output) neurons. However, I would like to create ...
0
votes
0answers
15 views

Use both historical prices and fundamentals data for predicting portfolio profit?

Get highly accurate probability distribution for the future price of portfolio using all the data available. Each stock has two historical data - daily prices (scalar time series updated daily) and ...
1
vote
0answers
16 views

Robust sum of non-independent random variables

What approach could be used to sum non-independent variables? I have probability distributions of stock prices and want to calculate the probability distribution of the portfolio price (sum of some ...
0
votes
1answer
31 views

Independence assumption of one-way ANOVA

Let's say that I am interested in how exam test score varies with a person's height. To do this, I take a sample from a school, and then I split this sample into two categories, (1) shorter than 165cm ...
0
votes
1answer
16 views

Range of probability for non-independent events

Suppose the probability of rain on day 1 is $p_1$ and probability of rain on day 2 is $p_2$. Then the probability of rain for the entire two-day period is $1-(1-p_1)(1-p_2)$, under the assumption that ...
0
votes
0answers
26 views

Does a martingale difference sequence (mds) imply strong mixing?

I read this from an econometrics paper (Dufour and Pelletier, 2014) " The typical hypothesis which is imposed in the time series literature is that the $u_t$'s are either independent and identically ...
1
vote
0answers
34 views

Independence of random variables and sums of random variables

I am seeking to find the joint distribution of X and Y. I have the marginal distributions of X and X+Y and they are independent. We have that $f(X=x,Y=y)=f(X=x,X+Y=x+y)$ which is equal to $f(X=x)f(X+...
0
votes
1answer
24 views

Using a sample obtained via McMc for Monte Carlo

Suppose we have a sample obtained via McMc. In order to use this sample for Monte-Carlo we need the sample to be independent. However our sample obtained via McMc is not independent, how can we ...
0
votes
1answer
16 views

How to conduct a test for independence in case of skewed classes (experiment design)?

The setting is as follows. We have a population of size $N$. Each subject has two properties $A$ and $B$, which can be either true or false. The question is: if for a random subject $A$ holds, is the ...
0
votes
0answers
7 views

What's the effect of observed variables on PGMs?

For a class we're going over the basics of PGMs. The below example was illustrated to show how (conditional) independence is somehow related to what variables are observed. This concept is perplexing ...
3
votes
3answers
236 views

Relationship between exchangeability and independence

If the random variables $X_1,...,X_n$ are I.I.d. when conditioned on an unknown parameter $\theta$, where $p(\theta)$ represents our prior beliefs about $\theta$, it follows from the commutativity of ...
0
votes
0answers
8 views

What's the latent random variable that explains Polya's urn?

According to De-Fenitte, exhangeable r.v's can be thought of as iid given a latent random variable. My question is: for a Polya's urn model, what would this hidden random variable be?
0
votes
0answers
9 views

Independence test or multinomial distribution test

I have a contingece table as follows: I can use the Chi-square test to check the independent of the column and row variable. You will compute the difference between the actual frequency and expected ...
0
votes
0answers
6 views

What would happen to a partial dependence plot if the assumption of independence was violated?

I understand the largest issue with PDPs is that it's assumed that the feature/s of the plot that are calculated are not correlated with other features. Let's use something like height and weight to ...
0
votes
0answers
21 views

Does random forest (and, decision tree) require an independent observation assumption? [duplicate]

I am wondering if random forest models require an independent observation assumption. My date includes observations from the same participants, but I do not have a way to identify each participant. ...
2
votes
0answers
19 views

Why is assuming Determinacy the same as Independence?

Consider a standard linear model $$y = x\beta_0 + \epsilon$$ or perhaps more generally a non-linear model $$ y = h(x,\beta) + \epsilon$$ I have several times come across the phrase that: To treat ...
0
votes
0answers
27 views

Independence of a Gaussian random variable and the product of another Gaussian random variable and a Bernoulli random variable

Let $X$ and $Y$ be two independent Gaussian random variables with mean $0$ and variance $σ^2_X$ and $σ^2_Y$ respectively. Let $Z$ be a random variable measurable with respect to $σ(Y)$ and suppose ...
0
votes
0answers
11 views

T test check requirements p-p plot [duplicate]

I am trying to perform a t-test on two independent groups and before doing that, I'm checking if the data meets the t-test requirements. One of the conditions is to test the assumption of normality. ...
2
votes
1answer
39 views

linear regression independence issue

I am not understanding why the predicted value and the true value are independent, I only know the error is independent of the predicted value. How do I proceed, thank you!
1
vote
0answers
18 views

Slight confusion on independent vs dependent samples

If I have two patient populations with cancer and I'm comparing cancer sizes, so say cancer size 1, 2 and 3 in an ordered fashion. Would these samples be dependent or independent? In one way, these ...
2
votes
4answers
124 views

which is the meaning of scatterplot between a pair of 2 consecutive pseudo random numbers with respect to the independence of the sequence?

Pseudo random number generators should give as output random sequences u1, u2, ... that are mutually independent and identically distribuited (iid). Since testing for independence is not easy, the ...
12
votes
3answers
962 views

Are independent variables necessarily “independent” and how does this relate to what's being predicted?

I'm fairly new to statistics. I'm not clear on the meaning of independent and dependent variables and the relationship to what's being predicted. In my text, as an example there is a data set ...
2
votes
1answer
52 views

Simple Question On Independent Events

I'm reading DeGroot and it says: A and B are independent if and only if Pr(A|B) = Pr(A) and Pr(B|A) = Pr(B). My question is: Do both equations need to hold for the two events to be independent ...
0
votes
0answers
30 views

Zero covariance and independence

Is there any other multivariate distribution $f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)$ that zero covariance between $X_i$ and $X_j$ implies independence between $X_i$ and $X_j$ besides multivariate ...
2
votes
1answer
25 views

Are these two distributions independent?

In a probability cheatsheet, there's the following claim: That $Z_1$ and $Z_2$ should be independent seems very unintuitive to me. It seems to me that if $Z_1$ is high, then $Z_2$ is low, and vice-...
1
vote
1answer
20 views

How randomly selected events are independent?

I've got confused while understanding the concept of independent events when cases are randomly selected: like the problems below. A poll finds that 72% of Jacksonville consider themselves football ...
0
votes
1answer
35 views

Interpreting the results of Chi-Square test

I have the following data: ...
0
votes
0answers
7 views

If outputs are independent why can we drop the condition on other outputs' inputs?

In a book the author is trying to explain why we cannot assume independence among outputs but rather conditionally independence. He gives this example: Imagine we had values of all Olympic years ...
0
votes
1answer
17 views

Posterior distribution and multiple parameters

In this problem I am doing I am trying to model count data with the negative binomial distribution. $k|(r,p)$ ~ $negBinom(r,p)$ Where we have the following priors: $ r $ ~ $Exp(2/3)$ and $p$ ~ $...
1
vote
1answer
50 views

Why aren't all Normal and Rademacher variables independent?

In Wikipedia page: https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent, we find a classical counter-example showing that two normally distributed and ...
0
votes
0answers
10 views

Finding probability of $U$ independent of joint pmf $f_(X,Y)(x,y)$ - meaning?

Given two random variables, $X$ and $Y$, with a joint pmf: $f_{(X,Y)}(x,y) =$ \begin{array}{ll} 0.3 ,\text{if} \space (x,y) = (3,0) \space \text{or} \space (0,6) \\ 0.2, \text{if} \space (...
0
votes
0answers
17 views

How to understand the normality assumption of the errors?

We know one of the Gauss-Markov conditions is $\mathrm{Cor}[e_i, e_j]=0$ for $i \neq j.$ Also, we assume the errors are normally distributed, i.e., $e_i \sim \mathcal{N}(0, \sigma^2).$ My teacher ...
2
votes
2answers
41 views

Testing independence of n binary variables

I have 6 binary (dummy) variables corresponding to different possible states that could apply to the subjects, and each subject has two of the six possible states, ie the sum of all of their dummy ...
0
votes
0answers
54 views

Correlation between two linear sums of several products of random variables

I would like to understand the correlation between two linear sums of several products of random variables. Well, it is quite difficult to describe the exact question I have. Basically I want to ...
4
votes
1answer
79 views

Characteristic function of $X+X^2$ if $X$ is a standard Gaussian random variable

Can anyone please tell me how I can find the characteristic function of the random variable given by the sum $X+X^2$ where $X$ is a standard Gaussian random variable? Many thanks!
1
vote
1answer
48 views

Are the random variables XY and X^2 independent if X and Y are zero - mean Gaussian independent random variables?

Assume that X and Y are random variables which are normally distributed as N(0,s_X^2) and N(0,s_Y^2), respectively. Furthermore, assume that X and Y are independent. Could anyone please tell me if ...
0
votes
0answers
45 views

In Simple Linear Regression $\hat \beta_1$ and $\bar Y$ are independent [duplicate]

I want to show that, in simple linear regression $\hat\beta_1 $ and $\bar Y$ are independent. My attempt: I have calculated the $\mathcal Cov(\hat \beta_1,\bar Y)$ and it turns out to be $0$.I also ...
14
votes
3answers
314 views

Where is the bomb: How to estimate the probability, given row and column totals?

This question is inspired by a mini-game from Pokemon Soulsilver: Imagine there are 15 bombs hidden on this 5x6 area (EDIT: maximum 1 bomb/cell): Now, how would you estimate the probability to find ...
0
votes
0answers
9 views

joint distribution of a random matrix and its column size variable

Suppose $C_k$ is a random matrix contains columns of measurement vectors that are random variables: $$C_k=[c_k^1,...,c_k^{m_k^c}]$$ $m^c_k$ is the number of columns as well as a random variable. All ...
1
vote
1answer
21 views

What defines a correct / incorrect model in Bayesian inference when it comes to independence

This might be a very broad question but I'm wondering whether we can say a model of Bayesian inference is "correct" or not about assuming independence. For example, suppose there are $N$ coins each ...
1
vote
1answer
43 views

Does the conditional density $f(X|X+Y<a)$ equal to $f(X)$ when $X,Y$ are independent continuous random variables with $Y$ having full support?

I wonder if the following equation holds when continuous random variables $X,Y$ are independent from each other and random variable $Y$ has full support on $\mathbb{R}$. \begin{align} f(X)=f(X|X+Y<...
2
votes
1answer
72 views

Probability of multiple parasites given the presence of one parasite

I am trying to simulate parasitic plants that infect trees. I would like to test wether the presence of one parasite makes it more likely that the tree has more than one parasite. The number of trees ...
9
votes
2answers
339 views

Independence of Mean and Variance of Discrete Uniform Distributions

In the comments below a post of mine, Glen_b and I were discussing how discrete distributions necessarily have dependent mean and variance. For a normal distribution it makes sense. If I tell you $\...
1
vote
0answers
34 views

Violation of Independence of Observations - Logistic Regression

Apologies if this seems a basic question but I can't find enough detailed information about what exactly constitutes a violation of independent observations. The regression I'm wanting to run is ...
1
vote
1answer
38 views

Do i need independence for paired t-test?

I am testing the mean change in IOP of eyes in a glaucoma study. When looking at just the 'Primary Eye', i am using a paired t-test to test the mean change from baseline as this is one eye per patient....
0
votes
2answers
19 views

Arch models: dependence and squared residuals

I would like a mathematical and intuitive answer to those questions: Why a dynamics like arch in the volatility of a time series implies that the time series is dependent (although not autocorrelated)...