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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Establishing independence between two random variables

I am currently working through some basic exercises in probability and have run into a snag. I am given two independent random variables $X$ and $Y$ that are both exponentially distributed with ...
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Show that the random variables $X$ and $Y$ are uncorrelated but not independent

Show that the random variables $X$ and $Y$ are uncorrelated but not independent The given joint density is $f(x,y)=1\;\; \text{for } \; -y<x<y \; \text{and } 0<y<1$, otherwise ...
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Does independence of two random variables imply uncorrelatedness?

There are many materials about the reverse question: "Does uncorrelatedness tell us something about independence?" But how to answer the question I've posed and why? Is there some simple ...
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Simplify $\int_{\Omega} \int_{\Omega} 1_{A}(\omega) g(\tilde{\omega}, \Pi(\omega)) d\mathbb{P}(\tilde{\omega}) d\mathbb{P}(\omega)$

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space and $D$ be a compact topological space, equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(D)$. Furthermore, ...
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Is Independence Stable under Intersections?

Let $A,B,C$ be events. If $A$ and $C$ are independent, and $B$ and $C$ are independent, does it then hold that $A\cap B$ is independent of $C$ as well?
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A Probability problem of a three-sided die with faces numbered 1, 2, and 3 from MIT 6.041

Problem is here 2(d) Original Solution is here 2(d) My approach: Let A be the event that at least one roll results in a 3 $$P(A)=1−P(no\ rolls\ resulted\ in\ 3)=1− (3/4)^6$$ Now let K be the random ...