Questions tagged [independence]

For questions involving the notion of independence of events, of independence of collections of events, or of independence of random variables. Use this tag along with (probability) or (probability-theory). Do not use for linear independence of vectors and such.

1,451 questions
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Expected distance between leaf nodes in a binary tree

Let T be a full binary tree with $8$ leaves. (A full binary tree has every level full). Suppose that two leaves a and b of T are chosen uniformly and independently at random. The expected value of the ...
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Statistics Independence Question: Given that an item has passed inspection, what is the probability that it is actually flawed?

The problem I have a question about is below. I only have a question on part (e), but I included the other parts of the question and answers as a reference. A quality control inspector is examining ...
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Independence of increasing limits

If $\{E_n\}_{n\ge1}$, and $\{F_n\}_{n\ge1}$ are increasing and independent for each $n$, show that their limits are independent. Here is my attempt: Note that $\{E_n \cap F_n\}$ is also increasing. ...
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Fill a joint table distribution, find covariance and check if two variable are independent.

Choose numbers from {2, 3} by tossing a fair coin; the coin is tossed twice. Choose 2 if the coin turns up Heads and 3 if the coin is Tails. So the possible outcomes are {(2, 2),(2, 3),(3, 2),(3, 3)}. ...
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Determining whether two events are independent or dependent.

I'm trying to make sure that my reasoning is correct for these problems. Say if the following pairs of events should be modeled as independent or dependent. Explain your reasoning. We choose a voter ...
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Expectation of ratio of normal and root chi-square

Let $X_1,X_2,X_3, X_4$ be i.i.d $N(0,1)$ random variables. What is the expectation of $$(X_1-X_2+X_3)/\sqrt {X_1^2+X_2^2+X_3^2+X_4^2}$$? I know how to obtain t-distribution but I wonder if the above ...
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Under what condition are $U$ and $V$ uncorrelated?

Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X + Y$ and $V = XY$. Under what condition are $U$ and $V$ uncorrelated? MY ATTEMPT We say that two variables are ...
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Independence Among Dataset Observations

The Machine Learning algorithm I would like to implement assumes that observations are obtained independently. What test could I perform in order to validate this assumption? Would that be a ...
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Consider of drawing one card from a deck of $52$. Prove that the events of a spade being drawn and an ace being drawn are independent events.

Consider of drawing one card from a deck of $52$. Prove that the events of a spade being drawn and an ace being drawn are independent events. Let $A$ be the event that a spade is drawn and let $B$ be ...
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Is the assumption of conditional independence fulfilled (based on 2D scatterplot)?

How do I find out based on a scatterplot, if the assumption of conditional independence is fulfilled? I'd be glad about example plots for the following cases: Case 1: categorical Y, two numerical ...
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What would be the expected product of two samples from same distribution?

What would be the $\mathbb{E}[x_ix_j]$ while $x_i,x_j \sim X$ where $x_i$ and $x_j$ are independent and X have finite moments.
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Independence of probability of $a_k$ being the largest element among the first $k$ elements in the permutation

The question is: Let $n \ge 2$ be an integer and consider a uniformly random permutation ($a_1$, $a_2$, . . . , $a_n$) of the set (1, 2, . . . , n). For each $k$ with $1 \le k \le n$, define the ...
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Prove convergence in distribution.

We have real-valued random variables $\{X_n\}_{n=1}^\infty$, $\{Y_n\}_{n=1}^\infty$, $X$ and $Y$. $X_n \rightarrow X$ in distribution and $Y_n \rightarrow Y$ in distribution, respectively. Also, $X$ ...
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Check if functions are independent

So I recently learned about how to check whether functions are independent. As far as I understood it one of the methods is to plug in freely chosen values for x and you can calculate the determinate ...
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Why is this sequence of random variables pairwise independent?

I have a sequence $(X_n: \Omega \to \mathbb{R})_{n=1}^\infty$ of pairwise independent random variables. Define for $n \geq 1: X_n' := X_n I_{\{X_n \leq n\}}$ where $I_A$ is the indicator function on ...
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Doesn't *identically distributed* imply *independent*?

What the title says. If I draw a random value $x_1 \sim \mathcal{N}(\mu, \sigma)$ a minute later, I draw another $x_2 \sim \mathcal{N}(\mu, \sigma)$ they come from identical distributions. Is there ...
Prove that $\mathbf{E}(Y|\sigma(X))=\mathbf{E}(Y|\sigma(X,Z))$
Let $Z$ be a random variable independent of $(X,Y)$. Prove that $\mathbf{E}(Y|\sigma(X))=\mathbf{E}(Y|\sigma(X,Z))$ My attempt: It is obvious that \$\int_A\mathbf{E}(Y|\sigma(X,Z))d\mathbf{P}=\int_A\...