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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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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1answer
28 views

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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1answer
32 views

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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1answer
39 views

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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1answer
65 views

Assume that $Z_i\sim\mathcal{N}(0,1)$ are independent and prove that $\sum\limits_{i=1}^{n}(Z_i-\overline{Z})^{2}\sim\chi^{2}_{(n)}$

Assume that $Z_{i}\sim\mathcal{N}(0,1)$ are independent and prove the following results (a) $\displaystyle\overline{Z} = \frac{1}{n}\sum_{i=1}^{n}Z_{i}\sim\mathcal{N}(0,1/n)$ (b) $\overline{Z}$ and $...
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2answers
78 views

How to find the expected value of the first order statistic

There are $Y_1, Y_2, \dots ,Y_n$ which are identically and independently distributed with pdf $4[(1-y)^3]$ for $0<y<1.$ We were asked to find the pdf of the first order statistic of which I got $...
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3answers
51 views

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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1answer
26 views

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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1answer
50 views

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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1answer
80 views

If $X$ standard normal and $Y$ symmetric Bernoulli are independent then $Y$ and $XY$ are independent?

If $X$ standard normal and $Y$ symmetric Bernoulli are independent then $Y$ and $Z=XY$ are independent? It is quite intuivively obvious that $Z\sim\mathcal N(0,1)$ and that would imply that $Y$ and $...
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2answers
322 views

If the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?

I know that if $\mathbb{E}[X]=\mathbb{E}[X|Y] , \mathbb{E}[Y]=\mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle. Now we add the variance, if $$\...
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1answer
18 views

short question about independence of random variables (specific example )

Okey imagine that you have two random variables $ X,Y \sim Bin(1,\frac{1}{2})$, which are independent. Define $Z_1, Z_2$ with $ Z_1 := | X - Y | $ and $Z_2 = X + Y$. Show that $Z_1$ and $Z_2$ are not ...
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1answer
22 views

Linear independence proof of sublist from a list of dependent vectors

Let $\lambda$ be an eigenvalue of $A$, such that no eigenvector of $A$ associated with $\lambda$ has a zero entry. Then prove that every list of $n-1$ columns of $A-\lambda I$ is linearly independent. ...
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1answer
34 views

Confusion about Independent events

For two events $E$ and $F$, we have $P(E)$ and $P(F)$ as the respective probabilities of there occurrence. Denote $P(EF)$ as the probability of their simultaneous occurrence. If $P(EF) = P(E) \cdot P(...
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2answers
32 views

Unclear theorem about vector spaces

Can anyone explain the following theorem? When I try to understand it, it seems like a contradiction of a basis. Is the vector v mentioned in the theorem not a ...
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1answer
39 views

Does matrix multiplication of a random vector preserve independence?

Let $\boldsymbol{X},\boldsymbol{Y} \in \mathbb{R}^n$ be random vectors and let $\boldsymbol{A} \in \mathbb{R}^{m \times n}$ be a non-square matrix of constants i.e. m < n. Suppose that each element ...
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1answer
60 views

relation between inquisitive logic and logic as games?

In the very intriguing thesis "Questions in Logic" Ivano A. Ciardelli shows how to build a semantics of questions that reduces to Truth Conditional logic for factual statements where ¬¬p = p, but has ...
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1answer
25 views

Are the row vectors in a row reduced echelon matrix always independent?

Are the row vectors in a row reduced echelon matrix always independent? I'm thinking that since the first row is the only row with a non-zero coefficient, then it must be independent of all the ...
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1answer
56 views

Show $Z_{n} \xrightarrow{d} \mathcal{N}(0,1)$

Let $(X_{k})_{k \in \mathbb N}$ a sequence of independent random variables and $F_{k}=F_{X_{k}}$ the respective cdf functions of $(X_{k})_{k \in \mathbb N}$ that are both continuous and strictly ...
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1answer
43 views

Hints find $Z$ so that $n \min\{X_{1},…,X_{n}\}\xrightarrow{d} Z$

Let $n \in \mathbb N$ while $(X_{i})_{i=1}^{n}$ are independent and uniformly distributed random variables on $[0,1]$ and define $M_{n}:=\min\{X_{1},...,X_{n}\}$. Find a $Z$ so that $n M_{n}\...
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1answer
45 views

Independence system/ graph-theory

$ (E,S)$ is an independence system with $ w: E \rightarrow \mathbb{R_+}$. $E= \{ e_1,...,e_m\} $ is the set of edges with $ w(e_1) \geq....\geq w(e_m), w(e_{m+1}):=0$. Define the set: $ E_i:= \{ e_1,....
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1answer
75 views

Show that $\max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9}$

Let $A$ and $B$ be independent events. Show that \begin{align*} \max\{\textbf{P}((A\cup B)^{c}),\textbf{P}(A\cap B),\textbf{P}(A\triangle B)\}\geq\frac{4}{9} \end{align*} MY ATTEMPT Since $\textbf{P}...
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0answers
76 views

Show that stochastic integral is Gaussian by independency of increments

Let $h \in L^2[(0,1)]$ and consider the process $(X_t)_{t \in [0,1]} = \big( \int_0^t h(s) \text{d}B_s \big)_{t \in [0,1]}$, where $B_s$ is Brownian motion. By construction of the integral I know ...
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2answers
58 views

$X_{1},…,X_{n}$ ~ $Ber(p)$ what can I say about $\exp(\lambda X_{1}),…,\exp(\lambda X_{n})$

Let $X_{1},...,X_{n}$ ~ $Ber(p)$ be independent, and $\lambda > 0$. What can I say about $\exp(\lambda X_{1}),...,\exp(\lambda X_{n})$ with respect to $\mathbb E[\exp(\lambda X_{i})]$? Well ...
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1answer
32 views

Show that $X \perp\!\!\!\perp Y \iff \forall f \, \text{bounded and measurable} \, \mathbb{E}[f(X)\mid Y] = E[f(X)]$

I could only prove one direction, let $A \in \sigma(Y)$ $$\int_A\mathbb{E}[f(X)\mid Y] d\mathbb{P} =\int_Af(X) d\mathbb{P} = \mathbb{E}[f(X) \mathbb{1}_A] = \mathbb{E}[f(X)] \mathbb{E}[ \mathbb{1}...
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1answer
26 views

Equivalent condition for a stochastic process to be independent of a $\sigma$-algebra [closed]

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $\Omega$ $I$ be a set $(E_i,\mathcal E_i)$ be a measurable space for $i\in I$,...
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1answer
36 views

Hints on proving existence of $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$

Let $(X_{n})_{n}$ be independent random variables that are $\mathcal{U}{[1,2]}$ Prove $(\prod_{i=1}^{n}X_{n})^{\frac{1}{n}}$ exists for $n \to \infty$ and that $\exists c \in \mathbb R$ such that $(...
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1answer
34 views

Independent continuous random variables problem

I have some problem at how to determine if two random variables are independent or not. If X and Y are two continuous random variables and their join probability density function is listed below $$ f(...
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2answers
39 views

$\sum_{n=1}^{\infty}p_{n}<\infty \iff X_{n} \to 0$ a.s.

Let $(p_{n})_{n}\subset [0,1]$ and $(X_{n})_{n}$ independent random variables, so that $X_{n}$~ $Ber(p_{n})$ Prove that: $\sum_{n=1}^{\infty}p_{n}<\infty \iff X_{n} \to 0$ a.s Ideas: "$\...
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0answers
18 views

Sequence of log-normal distributed random variables

Let $X_1, X_2..$ be a sequence of independent log-normally distributed random variables. Prove that there exists a constant $c\in\mathbb{R} $ such that $$\lim_{n \to \infty}\sqrt[n]{\prod_{i=1}^{n}X_i}...
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0answers
16 views

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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1answer
30 views

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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15 views

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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1answer
38 views

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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34 views

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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1answer
32 views

$2N$ independent random variables. Does the sum of $N$ random variables and the sum of the other $N$ are independent?

$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$ I'm quite sure that it's true, but I don't really know how to prove that.
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1answer
20 views

Probability of transmitting a signal through a network of transmitters.

We have a network of four transmitters $A$, $B$, $C$ and $D$. What is the probability of transmitting a signal through the network if all transmitters work independently and the probabilities of ...
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Which scenario gives you more chances to win, assuming that the results of each match are independent?

You are to play three matches, and you have to win two consecutive matches our of those three. There are two scenarios in which you can play: against a champion, against a friend, then against a ...
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128 views

show that $I, T, T^2, …, T^k$ are linear dependent

I am learning linear algebra and new to it. I can not solve this problem. I think it has a trick that I don't know. for T(a linear map), $T:V\rightarrow V$ and every $v$ in $V$ the $v, T(v), T^2(v),...
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0answers
33 views

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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1answer
48 views

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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2answers
45 views

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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33 views

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 ...
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0answers
29 views

Meaning of “Any two deterministic quantities are independent”

I'm having trouble understanding this statement: "any two deterministic quantities are independent" The example the text provides is as follows: $$Prob(\varnothing\cap\Omega) = Prob(\varnothing) = 0 ...
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1answer
40 views

Equivalence of the condition that the supremum of i.i.d. RVs are finite a.s.

I am proving the following : Suppose $\{X_n : n\in\mathbb{N}\}$ are i.i.d. random variables. Then $P(\sup_{n\in\mathbb{N}}X_n < \infty) = 1$ if and only if $ \sum_{n\in\mathbb{N}}{P(X_n > M)} &...
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2answers
87 views

What can be said about i.i.d. $X$ and $Y$ such that $XY=(X+Y)/2$ in distribution?

Let $X$ and $Y$ be i.i.d. If $(X+Y)/2$ is equal in distribution to $XY$, then what do we know about the distributions of $X$ and $Y$? I feel like I can't say much about these distributions. I can ...
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0answers
12 views

Lemma relating linear independence to subspaces. How? (Suppose x…xs are vectors which span W and y…yt are independent vectors in W. Then s>=t)

Sorry for the absolutely horrid title, didn't want to have a vague one. Here's the full lemma: Let W$\subset R^n$ be a subspace Suppose that ${u_1,u_2}$...${u_s}$ are vectors which span W, and ${v_1,...
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1answer
20 views

Unclear step when proving linear independence

I don't understand how we choose the x-values to solve the equation system proving linear independence. For example I have this question: Proving the linear dependence is trivial, but then this ...
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0answers
33 views

Prove if two variables are conditionally independent

I have 4 random variables A, B, C, D. I know that the joint is $$p(A,B,C,D) = p(A)p(B|C,A)p(C)p(D|B,C) $$ And I want to prove (if true) that $$ A \perp D | B $$ I have tried this: $$p(A,D,B) = \sum_C ...
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1answer
28 views

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\...