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Questions tagged [probability-theory]

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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9 views

A question about the proof of Theorem 5.21 in Van der Vaar(1998)

If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \begin{equation} \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\...
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0answers
29 views

“expectation of sum is sum of expectation”, is this claim true? if yes, how to justify this claim?

this post is saying linearity of expectation gives following equation $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$ per wiki, Linearity of Expected_value is ...
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1answer
11 views

For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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0answers
6 views

Intuition of TailVaR

As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L. I don't really get that, so I tried splitting the equation into: $...
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1answer
9 views

Sampling with replacement. $P[\upsilon_m>n]=\prod_{i=2}^{n}(1-\frac{i-1}{m})$.

Let $\{X_n,n \leq 1\}$ be i.i.d. and uniformly distributed on the set $\{1,...,m\}$. In repeated sampling, let $v_m$ be the time of the first coincidence; that is, the when we first get a repeated ...
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0answers
13 views

a game on compact subset of reals

Suppose $g_1,\dots, g_5: S\to S$ are non-trivial, continuous functions on S , a compact subet of real, are selected at random with probability measure $p$ on $\{1,2,\dots, 5\}$ and the selection is ...
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1answer
12 views

Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $...
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30 views

Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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1answer
26 views

Expected value of the number of different numbers drawn in 37 rounds of roulette?

I need help with this problem. What is the expected value of the number of different numbers drawn in 37 rounds of roulette? Is this possible to interpret as the number of records? So the expected ...
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17 views

Proving $h(X_n)\implies h(X)$ in distribution

Let $X_n$ be a sequence of random variables on $(\Omega,\mathscr{F},P)$such that $X_n\implies X$ (converges in distribution). Let $h:\mathbb{R}\to\mathbb{R}$ be a function whose points of ...
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2answers
37 views

$X_i$, $i=1,2,..n$ independent R.Vs $P(X_i=1)=\frac{3}{4} ,\ P(X_i=-1)=\frac{1}{4}$. Prove $\sum_{i=0}^nX_i \to \infty$ a. s. as $n \to \infty$

I am asked to prove $X_1+X_2+X_3+...+X_n$ diverges almost surely as $n \to \infty$ Let $Y_n=X_1+X_2+...+X_n$ then what we want to prove is $P(Y_n=k)=1, \text{ as} (k,n) \to (\infty,\infty)$ Let us ...
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1answer
16 views

Probabilistic PCA: Derive $\mathbb{E}[z_n \mid x_n]$

Consider the probalistic pca setting, where $x \in \mathbb{R^d}$ is an input vector drawn from $p(x)$, $z \in \mathbb{R^m}$ is an explicit latent variable with $p(z) = \mathcal{N}(0,\mathbb{I}_m)$ ...
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0answers
17 views

How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
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1answer
16 views

Distinct random variables in sampling

Let $X_1,X_2,\dots$ be i.i.d. random variables with $X_1 \sim U[0,1]$. Throwing out a null set all the variables are distinct. Can anyone explain this second sentence? What does he mean with "...
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1answer
26 views

It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $\Bbb N$ because there is no constant $c\in\Bbb R$ such that $\sum_{k=1}^\infty c=1$. Using ...
4
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2answers
47 views

If $Y\sim\operatorname{Beta}(a,1-a)$ and $Z\sim\operatorname{Exp}(1)$, then $YZ\sim\operatorname{Gamma}(0,1)$?

I have two random variables $Y \sim \operatorname{Beta}(a, 1 - a)$ $Z \sim \operatorname{Exp}(1)$ If $Y$ and $Z$ are independent, why is the distribution of $X = YZ \sim \operatorname{Gamma}(...
2
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1answer
13 views

Actual meaning of the formula $E[\phi(X,Y)|\mathcal{G}]= E[\phi(x, \cdot)]|_{x=X}$

Let's suppose we have a probability space $(\Omega,\mathcal{F},P)$ and a sub sigma algebra $\mathcal{G} \subset \mathcal{F}$. Let's say we have $X$ that is $\mathcal{G}$-measurable and $Y$ that is $\...
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0answers
19 views

Strong Markov property and another stopping time

I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the ...
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0answers
22 views

Tricky information inequality $I(X;Z) \geq H(T)$

I am wondering whether $I(X; Z) \geq H(T)$ when the following conditions hold: $H(T | X) = H(T)$ $H(T | Y) = H(T)$ $H(T | X, Y) = 0$ $H(Y | Z) = H(T | Z) = 0$ $X, Y, Z, T$ are discrete. I know first ...
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1answer
23 views

about Random Variables Expectation

suppose $\mathcal{G} \space is \space \mathcal{F}'s \space sub \space \sigma-algebra, and \space X\in\mathcal{L^1}(\Omega, \mathcal{F}, P), \space\ $$\ Y\in\mathcal{L^1}(\Omega, \mathcal{G}, P)\ .$ ...
0
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1answer
15 views

Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
2
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1answer
54 views

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
0
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1answer
32 views

What is the PMF of the product of two discrete independent random varibales? [on hold]

I would like know how can I multiply two independent discrete random variables. Let $X$, $Y$ be two discrete independent random variables. What is the probability mass function of $Z=X Y$? For two ...
0
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0answers
24 views

Show that a càdlàg function is uniformly right-continuous on compact intervals

Let $(E,d)$ be a locally compact separable metric space, $I\subseteq\mathbb R$ be an interval, $f:I\to E$ be càdlàg and $a,b\in I$ with $a<b$. How can we show that $\left.f\right|_{[a,\:b]}$ is ...
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1answer
23 views

Conditional Independence and product of random variables

I am stuck at the following situation: Let random variables $Y, X, W_1, W_2$. I know that $W_1$ and $W_2$ are each independent from $Y$ conditional on $X$: $$p\left(Y\mid \{X,W_1\}\right) = p\left(...
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0answers
20 views

Find a distribution. Wiener proccess [on hold]

Find the distribution of $$\frac{1}{t-s} \left(W_t^2 + W_s \left[ \frac{t}{s} W_s - 2W_t \right] \right), \qquad 0 < s <t. $$ How do i do this? Where $$ W_t, W_s $$ - Wiener process
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1answer
21 views

Is $x \mapsto P_x(A)$ measurable for a measure $P_x$ determined by transition kernels $(\delta_x,P_i)_{i \ge 0}$

Given a sequence $P_i$ of transition kernels from $(E,\mathcal B(E)$ to $(E,\mathcal B(E))$ and $\delta_x$ the Dirac delta measure for the point $x\in E$, it follows from the Ionescu-Tulcea theorem ...
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1answer
50 views

If $f\in C_0$ and $\lambda>0$, how can we show that $x\mapsto\int_0^\infty e^{-\lambda t}f(x(t))\:{\rm d}t$ is continuous wrt the Skorohod topology?

Let $(E,d)$ be a locally compact separable metric space, $C_0(E)$ denote the space of continuous function from $E$ to $\mathbb R$ vanishing at infinity equipped with the supremum norm and $D([0,\infty)...
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1answer
11 views

Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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0answers
16 views

calculate the probability of error in a array of bits

I need to calculate the probability in a certain problem. So there are 555 random bits [1 0 1 0 ... 1 0 0 1 1]. These 555 bits are divided in 37 parts of 15 bits each. Of the 555 bits, X bits flip at ...
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0answers
9 views

When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
0
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0answers
17 views

Covariance of two random variables, linear relationship and normalization of covariance

The covariance of two random variables $X$ and $Y$ is given by $$\displaystyle\operatorname{cov}\left[X,Y\right]=\mathbb{E}\left[\left(X-\mathbb{E}\left[X\right]\right)\left(Y-\mathbb{E}\left[Y\right]\...
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73 views
+300

If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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1answer
31 views

On the convergence in probability of a sequence of random variables.

Let $\{X_t \}_{t \in \mathbb{N}}$ be a sequence of independent random variables such that $E[X_t] = \theta E[X_{t-1}]$ for all $t \in \mathbb{N}$ where $|\theta|< 1$ and $E[X_0] = \mu > 0$. ...
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0answers
20 views

justification of $\sum_{j\neq i}\mathbb{E}[Y_i Y_j] = (n - 1)\mu^2$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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2answers
27 views

which rule or definition apply ${\displaystyle {\begin{aligned} \operatorname {E} [Y_{i}^{2}] = (\sigma ^{2}+\mu ^{2}) \quad (3.1) \end{aligned}}}$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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0answers
36 views

Is every probability measure a Radon measure? [on hold]

Let $\mu$ be a probability measure defined on a compact convex subset $K$ of a locally convex Hausdorff space $X$. Is $\mu$ a Radon measure?
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0answers
54 views

Showing $E(X) = \sum_{i}E(X\mid A_i)P(A_i)$

Following is my proof. Suppose $X$ is a discrete-type random variable ranging in the set $S$ and $\{A_i : i=1,2,3,\dots\}$ is a finite or countably infinite partition of a sample space $\Omega$. We ...
0
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1answer
23 views

Uniform distributed success probability for a coin

$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
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0answers
12 views

Upper bound on number of cliques in a Vietoris-Rips complex

Does there exist an upper bound on the number of cliques of order $k$ in a Vietoris-Rips complex? I found this work --> https://arxiv.org/pdf/1104.0914.pdf I understand it makes the assumption of ...
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1answer
26 views

Odd moment and the characteristic function of a random variable

Let $X$ be a random variable and $\phi_X(t)$ be its characteristic function. Let $n$ be a positive even integer. If $\phi_X(t)$ is $n$-times differentiable, then the $n$-th moment of $X$ exists and ...
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2answers
35 views

What will be the probablilty in these cases? [on hold]

So we have a fair and unbiased dice, which is rolled thrice in a row. 1)What is the probability to get the sequence [1,2,3] in the three continuous trials? 2)What is the probability of getting the ...
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2answers
18 views

Finding the expected number of a certain colored ball drawn from an urn in k draws

Suppose we have an urn containing c yellow balls and d green balls. We draw k balls, without replacement, from the urn. Find the expected number of yellow balls drawn. Hint: Write the number of ...
0
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2answers
22 views

How is P(B) derived and why is $P(D_i)$ equal to 55/72 and not $(55/72)^i$

So this is a question with its solution below to which I don't understand 2 things. How is P(B) derived? And, why is $P(D_i)$=55/72 and not $(55/72)^i$. Since, for example, obtaining heads in the n ...
1
vote
1answer
21 views

$P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
0
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0answers
32 views

Is this application of a law of large numbers rigorous in this not identically distributed case?

Let $ \{ X_t \}_{t \in \mathbb{N} }$ be a sequence of indipendent random variables such that $X_t \sim N(u_t, 1)$ for all $t$ where the mean $u_t$ is given by the equation $$u_t = \theta u_{t-1} + \...
0
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0answers
15 views

Variance of linear combination

This is a follow up question to this. Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)...
1
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1answer
41 views

$m$ minimizes $E(|X-a|)$ over $a\in R$ if and only if $m$ is a median for $X$.

I'm trying to show that $E(|X-a|)$ attains its minimum value if and only if $a=m$ where $m$ is the median. I know this particular problem has been discussed before, but I want to prove it in a ...
1
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0answers
23 views

Quadratic martingale bound

I know that if $a_1,a_2,\dots$ are random variables and {$\mathcal{F}_{t}$ } is a filtration such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$, then for any stopping time $\...
0
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0answers
39 views

Understanding i.i.d. random variables from product measure space perspective

I have a very weak background in measure theory, and I am having some troubles understanding i.i.d. random variables from a measure theoretic perspective. Let $X$ be a random variable defined on a ...