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.

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Coin with probability p (Biased?)

\begin{array}{l}{\text { Suppose a coin is tossed three times independently, with probability of land- }} \\ {\text { ing heads } 0 \leq p \leq 1 \text { and a complement probability } 1-p \text { of ...
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Conditional Expectation on Several Variables [on hold]

The question is from a note of econometrics. For a linear model $Y_t=X_t'\beta +\varepsilon_t$: Assumption 1: {$Y_t,X_t'$}$_{t=1}^m$ is an i.i.d random sample. Assumption 2: $E(\varepsilon_t|X_t)=0$....
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Probability - Defective product

A company sends 30% of its product to Client A and 70% to Client B. Client A reports that 5% of the products it received are defective, whereas Client B reports that 4% of products received are ...
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Are $A|B$ and $B$ independent events? [on hold]

Suppose $A$ and $B$ are two dependent events, that is $P(A\cap B)>0$. We know that $P(A\cap B)=P(A|B)P(B)$. Is it true that $A|B$ and $B$ are independent? From my understanding, two events $X$ ...
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Show that if $X_1, \dots, X_n$ are i.i.d., then two expectations are equal.

Let $X_1, \dots, X_n$ i.i.d random variables. Put $S_n:= \sum_{k=1}^n X_k$. Show that $\mathbb{E}[X_1 I_{\{S_n \in A\}}]= \mathbb{E}[X_j I_{\{S_n \in A\}}]$ for $1 \leq j \leq n$, where $A$ is an ...
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If the probability of current flowing in circuit is known, how can I know the probability that a certain bulb will work?

Here the schema is very important: The probability that a bulb will work is 0,5. The probability that the current will flow in circuit is 0,3984375. What is the probability that the bulb C will work....
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What is the probability that the current will flow in the circuit, if P(1/2, that the bulb will work) and there are 7 bulbs.

So, the problem is as follows: Calculate the probability that the current will flow in circuit if the chance that a light bulb will work is 0,5 and there are totally 7 bulbs. Here is the schema: I ...
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$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0$

Let $\{A_n\}$ be a sequence of independent events. How to prove that $$P(\limsup A_n)=1 \Leftrightarrow \sum_{n=1}^{\infty} P(A \cap A_n) = \infty\; \forall A, P(A)>0?$$ As the $A_n$ are ...
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Solution of linearly dependent functions

I'm having a lot of trouble with this question. I know they are not linearly independent, but I'm not sure how to proceed. Here is the problem: Thank you.
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Independence of a Random Variable from a Sigma Field and Expectation

The question is: "Show that $X$ is independent of $\sigma(Y)$ if and only if for bounded and measurable $f,g$, $E[f(X)g(Y)]=E[f(X)]E[g(Y)]$." I think I have managed to prove the forward statement by ...
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Linear dependence of functions involving $e^x$

On a test today I was given the following functions: $$f(x) = (-1+x)e^x$$ $$g(x)=-2e^x$$ We were asked to show if it was linearly dependent or linearly independent So I showed that if I ...
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Calculating the expected value of the amount of same numbers chosen by two people.

so I've been stuck all day on one question and I have no idea what to. This is the problem: Two people choose from a set of integers ranging from 2 to 100.(so 99 different integers) One person ...
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Independent Events and Bernstein Paradox for n events [closed]

Is it possible to extend Bernstein Paradox example (about pairwise independence, but joint dependence of 3 events (color sides of tetrahedron)) to n events using the same reasoning?
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Are $A,B,C$ independent given that $P((A \cap B )\cup C)=P(A)\cdot P(B)\cdot P(C)$

Can someone help me shading light on this question about independence?The answers look conflicting. Are these 3 events independent? Not Solved yet. Can anyone help?
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Does converse implication for PGF of sum of independent random variables hold?

Let $X_1, X_2$ be i.i.d. random variables. Then the probability generating function $G_{X_1+X_2} = G_{X_1} G_{X_2}$. Does the converse implication hold? It is quite easy to check that it does for ...
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Independence of Brownian Motion and $\mathscr{F}_{0}$

Assume $B=\left\{ B_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ is a standard 1-dimensional Brownian motion. Then show that $\mathscr{F}_{\infty}^{X}$ and $\mathscr{F}_{0}$ are independent (...
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Independence with single random variables implies the independence of collection.

On probability space $\left(\Omega,\mathscr{F},P\right)$, assume a sub $\sigma$-algebra $\mathscr{F}_{0}$ and random variable $X,Y$ are independent with $\mathscr{F}_{0}$. Do we have $\left(X,Y\right)$...
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Conditional Independence for XOR

Let's say we have a basic XOR table with input variables X and Y and output Z. Could I assume that X and Y are conditionally independent? I think so, because by knowing Z, if I knew X I would also ...
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Interchange between expected value and infinite summation (Fubini theorem)

Let $S_n = \sum_{i=1}^nX_i$ (where the $X_i$ are i.i.d.) and let N be a positive, integer valued r.v., independent from the sequence $X_n$. Suppose also that $E[N]<\infty$ and $E[|X_i|]<\infty$. ...
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Independence between events involving three random variables

Let $X,Y,Z$ be three independent random variables, we want to find out if the following holds: $$P(X\geq Y,X\geq Z) = P(X\geq Y)P(X\geq Z)$$ that is, if $X\geq Y$ and $X\geq Z$ are independent events. ...