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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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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
46 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
40 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
32 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
69 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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23 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
32 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
29 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
18 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
32 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
26 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
27 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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16 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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15 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
28 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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0answers
7 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
35 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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2answers
32 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
24 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
16 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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45 views

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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2answers
123 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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24 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
45 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
29 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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26 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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21 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
35 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
84 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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8 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
14 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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23 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
24 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\...
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2answers
30 views

An independent collection, where the pairwise products are also independent

Can you give me a non-trivial example of a sequence $\{X_k\}_{k=1}^n$ of independent, and identically distributed random variables, such that, the pairwise products, $\{X_kX_{\ell}\}_{1\leq k<\ell \...
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29 views

Probability with current chain

I have this chain. $A,B,C,D,E $ are switches. They are independent from each other. They can be switched on with probability $p$ or switched off with probability $1-p$. We need to find probability ...
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1answer
26 views

How to show Sub-independent Random Variables are uncorrelated.

I want to prove the following: If two RVs $X, Y$ are sub-independent, i.e., $\phi_{X+Y}(t) = \phi_X(t)\phi_Y(t), t\in\mathbb{R}$ then $X, Y$are uncorrelated. Keep $Cov(X,Y) = E(XY)-E(X)E(Y) = 0$ in ...
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1answer
22 views

Prooving the Independence of two events

Let $A$ be event and probability $\mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event. So I think I need to proove $\...
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59 views

What will be the pdf of $X+Y$ if $X$ and $Y$ are iid from Cauchy? [duplicate]

Suppose $X$ and $Y$ follow Cauchy distribution independent of each other. What will be the pdf of $X+Y$? What I got by using convolution theorem is that the density $g$ of $X+Y$ is $:$ $$g(x) = \...
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1answer
32 views

What will be the pdf of $X+Y$?

Suppose $X$ and $Y$ are independent random variables. Let $f$ and $g$ be the pdf of $X$ and $Y$ respectively. Let $h$ be the pdf of $X+Y$ then can we say that $h(x)=f(x)g(x),$ for all $x \in \Bbb R$? ...
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14 views

Equivalence of independence and a conditional expectation equality

Let $E_1$, $E_2$ be Polish, let $D_{E_2}([0,\infty])$ be the space of cadlag functions with values in $E_2$ and set $\mathcal{F}_t^Y$ to be the $\sigma$-algebra generated by $Y_s$, $s \leq t$, ...
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1answer
62 views

Independence between random vector and event

Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables. Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables. Consider that $X_i$ ...
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2answers
152 views

Quadratic Formula With Independent and Dependent Variables

Given the differential equation $dy/dt = (y + t)^2$, we can apply the u-substitution $u = y + t$ to arrive at the separable differential equation $du/dt = u^2 + 1$. This separates to $1/(u^2 + 1)\ du ...
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0answers
21 views

How to show that $\sum_{i=1}^m (X_i−X_m)^2$ and $\sum_{i=1}^n(Y_i− Y_n)^2$ are independent

Let $X_1,...,X_m$ be i.i.d. sample with $N(\mu_1,\sigma^2)$, and $Y_1,...,Y_n$ be i.i.d. sample with $N(\mu_2,2\sigma^2)$. Let $S_x^2 = \sum_{i=1}^m (X_i−X_m)^2$ and $S_y^2= \sum_{i=1}^n(Y_i− Y_n)^2$...
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1answer
13 views

Seating Arrangement and Independent Events?

PROBLEM: Alice, Bob, Catherine, Doug, and Edna are randomly assigned seats at a circular table in a perfectly circular room. Assume that rotations of the table do not matter, so there are exactly 24 ...
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1answer
39 views

Proving linear independence and that $\dim V$ is greater than or equals 3.

Let $V= \operatorname{span}\{v_1,v_2,v_3,v_4\}$ be a vector space such that $v_i$ are unit vectors for all $i$ and $v_i.v_j<0$ if $i$ does not equals to $j$. (i) Show that no two vectors among {$...
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1answer
20 views

Discrete Probability: Determine if the events are Independent or not

Question: You flip a fair coin four times; these four flips are independent. De fine the events: A = "the first two flips result (in this order) in HT", B = "the second and third flips result in ...
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1answer
82 views

Given joint moment generating function, what value of $a$ makes $X + 2Y$ and $2X − Y$ independent?

I am new to joint moment generating functions and their properties, so am a bit stuck on how to begin the following problem: Given $M_{X,Y}(t,u) = exp[2t+3u+t^2+atu+2u^2]$, what value of $a$ makes $X ...
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0answers
29 views

Proof of a martingale

Consider a martingale $X_{n}, n \in \mathbb{N} $ with independent increments. Assume that the variance, $\sigma^{2},$ of the increments is constant. Define $Y_{n}=X_{n}^{2}-n\sigma^{2}.$ Prove ...
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1answer
33 views

Suppose $X_n \rightarrow X$ a.s. and for each $n$, $X_n \perp \textit F$. Then is it true that $X \perp \textit F$?

For random variables $X_n$ and $X$, suppose $X_n \rightarrow X$ a.s. and for each $n$, $X_n \perp \mathcal F$ i.e. independent with $\mathcal F$. My question is: Is it true that $X \perp \mathcal ...
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2answers
68 views

PDF of Z = XY for Jointly Uniform (X,Y) with Parabolic Region

"Suppose that $(X,Y)$ is uniformly distributed on the subset of $\; \mathbb R^2$ defined by the inequalities $0 < X < 1$ and $0 < Y < X^2$. Determine the probability density function of ...