All Questions
Tagged with probability independence
1,541
questions
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independence with respect to the sum of independent random variables
Assume we have $X$, $Y$, $Z$ are jointly independent random variables. Next, assume that
$$
X = X_{1} + X_{2} + X_{3}
$$
and $X_{1}$, $X_{2}$ and $X_{3}$ are jointly independent. Does it follow that
$...
- 119
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Probability that at least one event happens (dependent events)
Problem description
Assume we have a bag filled with marbles of two different colors, red and blue. Our goal is to be able to pick out at least one red marble by picking out the least amount of ...
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2
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How to directly calculate $P(A)^{\prime}$?
Six different coloured balls are placed in a box. Kendra and Abdul each select a ball without replacement. What is the probability that Kendra does not select the green ball and Abdul does not select ...
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Defining random variable on given sample space
Let $(\Omega, F, P)$ be a probability space with $\Omega = \{1,2,\ldots, 26\}$ and $P$ has uniform distribution
the task is to define two discrete independent random variables $X,Y$ with $X$ being a ...
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3
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Unions & Intersection of Probability
I've had all three answers below marked wrong, and I am not sure how to proceed. I have included my thinking.
Suppose we are interested in the buying habits of shoppers at a particular grocery store ...
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Simple ergodic convergence proof for iid
This is just a simple question from a problem sheet. Consider a sequence of independent identically distributed random variables $Y_0,Y_1,\dots$. Let $f$ be a function such that $\mathbb{E}|f(Y_0)|^2 &...
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43
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How to force a product of two i.i.d. random variable to be gaussian
This question is related to this other question of mine: I realized that my original question was maybe too abitious, and I would like to discuss a much more limited version of it.
Consider two real ...
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Independence of two argmax
Problem: Suppose I have two random variables $X_1$ and $X_2$, also I have a measurable (for each $\theta$) function $f(X, \theta)$, where $\theta$ is a scalar parameter. I know that for each $\theta \...
- 159
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Question about the Definition of Conditional Independence
For three random variables, $X$, $Y$, and $Z$, and they all have probability densities. We say that $X$ and $Y$ are considered conditionally independent given $Z$, $(X\perp Y|Z)$ if
$$
p(y|x,z)=p(y|z)
...
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Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf
Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
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Conditional Independence involving four events
I am trying to verify if $A \perp C | H$ and $B \perp C | H$ implies $A\cap B \perp C | H$.
So far, I did the below -- but feel I'm skipping something in the step indicated (*) below. Any help would ...
- 15
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1
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Independence of Sample Mean and Sample Variance
Let $X_1, \ldots, X_n$ be independent real random variables, with $n > 1$.
Define the sample mean by :
\begin{equation*}
\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i
\end{equation*}
Define the ...
- 2,468
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2
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Does Independence hold for $\sigma$-Algebras Generated by Disjoint Subsets of an independent Sequence
I want to show that for a sequence of independent random variables $(X_i)_{i \in \mathbb{N}}$ we have that for any two disjoint sets $A,B \subset \mathbb{N}$ we have that $ \sigma(X_i : i \in A)$ and ...
- 219
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Prove that $Z_n$ is independent of $(Y_{i,n})_i$
Suppose we have $Y_{i,n}, i \ge 1, n \ge 1 $ iid with expectation $\mu$. And given $Z_{n+1} = \sum_{i=1} ^{Z_n} Y_{i,n}$ and $Z_0 = 1$.
In lecture it was stated that $Y_{i,n}$ is independent of $Z_n$....
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X independent of Z and Y independent of Z imply XY independent of Z?
I have a question for which I have a somewhat unclear explanation.
Namely, if $X$ and $Y$ are each independent of $Z$ (i.e., $X$ is independent of $Z$ and $Y$ is independent of $Z$), then is $XY$ ...
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