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-...

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61 views

CLT question, the density function of the horizontal deviation from shot arrow to center is given…

The horizontal deviation from shot arrow to the center of the target is given as : $$ \varphi (x)= \begin{cases} 1- |x| , x\in (-1,1) \\ 0 , \text{ otherwise. } \end{cases}$$ If the horizontal ...
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0answers
23 views

Is something wrong with my confidence interval for a Binomial variable?

Let $X$ be a Binomial random variable with parameters $n$ and $p$. I came up with the following very simple approach to finding a confidence interval for $p$, but the Wiki page on confidence intervals ...
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1answer
19 views

Finding the probability density function for a series of RV - what am I doing wrong?

Let $X_1\sim Uni([0,1])$. We'll define a series of random variables - We'll choose a random point on the segment $[0,X_1]$ and call it $X_2$, then choose $x_3$ randomly on the segment $[0,...
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vote
1answer
44 views

Inverse Fourier-Stieltjes transform of $1$

Let $S(x) = \text{sgn}(x)/2$ for $x \ne 0$ and $S(x) = 0$ for $x = 0$. Then its Fourier-Stieltjes transform is $\hat{S}(k) = \int_{-\infty}^\infty e^{i k x} dS(x) = 1$. I tried to evaluate the ...
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0answers
66 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
0
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1answer
64 views

…Let $X_n \to X$ a.s and $E[X_n]=4$ [closed]

Let $X_n \to X$ a.s and $E[X_n]=4$, for $n = 1, 2, ...$ . Then always $E[X] = 4$.
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0answers
79 views

upper bound for $ \mathbb{E}[X] \cdot \mathbb{E}[1/X] $ [closed]

This is part of problem 2.6.46 in Shiryaev's 'problems in probability'. Let $X$ be a random variable which takes values in the interval $[a,b]$ for some $0<a<b<\infty$. Prove that $$ \mathbb{...
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0answers
20 views

Why does $1 \leq \sup \limits_{0\leq t \leq 1}( C|B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold?

Why does $1 \leq C\sup \limits_{0\leq t \leq 1}( |B_t|)$ P -as where $B$ is the standard B.M for some $C>0$ does not hold ? I am trying to show by contradiction that the Burkholder-Gundy ...
0
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1answer
42 views

Why is the probability that $3$ people will not respond to a survey equal to this?

When sent a questionnaire, $50\%$ of the recipients respond immediately. Of those who do not respond immediately, $40\%$ respond when sent a follow-up letter. If the questionnaire is sent to $4$ ...
1
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0answers
22 views

Deriving the CDF of sum of n iid random variables using the CDF of individual random variables?

If $Z=x_{1}+x_{2}+...x_{n}$ and all $x_i$ are iid then how can we find the CDF of $Z$ using the CDF's of individual random variables?
3
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2answers
159 views

In Borel-Cantelli lemma, what is the limit superior?

In a proof of the Borel-Cantelli lemma in the stochastic process textbook, the author used the following. $$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n} A_k$$ Can someone explain why lim ...
0
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0answers
55 views

find limit distribution by using central limit theorem.

$$x_1,...,x_n \sim \text{uniform (0,1)}$$ $$Y_n=\sum_i^n X_i$$ I want to find limit distribution by using central limit theorem. $E(Y_n)=n/2$ and $V(Y_n)=n/12$ And Moment generating function $M_{...
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0answers
39 views

Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-degenerate random variable.

Let $X_1,X_2,...X_n$ be iid with the same chi-square distribution with one degree of freedom. Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-...
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0answers
19 views

If $F_+^{-1}$ and $F_-^{-1}$ are the right- and left-cont. inverse of a CDF, resp., then $\left\{F_-^{-1}(y)<X\le F_+^{-1}(y)\right\}$ has prob. $0$

Let $X$ be a real-valued random variable and $$F(x):=\operatorname P[X\le x]\;\;\;\text{for }x\in\mathbb R\;.$$ Moreover, let $$F_-^{-1}(y):=\sup\left\{F\le y\right\}\;\;\;\text{and}\;\;\;F_+^{-1}(y):=...
0
votes
1answer
90 views

Convergence in probability : min and max forms of uniform distribution (0,1)

$X_i \sim \text{uniform(0,1)}$ independent variables. $Y_n = \min \{x_1,...,x_n\}$ $Z_n=\max\{x_1,...,x_n\}$ I want to show that both $Y_n$ and $Z_n$ respectively converges in probability to 0 and ...
2
votes
0answers
19 views

Distribution MSR

We have $Y_i = \beta_0 +\beta_1(X_i -\bar X )+\epsilon_i$ for i=1,...,n $$\epsilon_i \sim N(0,\sigma^2)$$ We know that $SSR= Y^T P_xY - n\bar Y^2=Y^T (P_x -n^{-1} J_nJ_n^T)Y$ $$J_n=\big[\begin{...
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0answers
21 views

Limiting distribution of the barycenter $\nu_n=\mathcal{N}\Big(\frac{1}{n}\sum_{i=1}^{n}m_i, (\frac{1}{n}\sum_{i=1}^{n}\sigma_{i})^2 \Big)$

I have a set of gaussian measures $\mu_1, \dots, \mu_n$ with $\mu_k=\mathcal{N}(m_k, \sigma_k)$. I am interested in the empirical Wasserstein Barycenter of this sequence which corresponds to $\nu_n=\...
0
votes
0answers
32 views

$X_n$ sequence of independent random variables, with distribution functions: $F_n(x)=\frac{n^x-1}{n-1}, 0\leq x \leq 1, n=1,2,3..$

$X_n$ sequence of independent random variables, with distribution functions: $$F_n(x)=\frac{n^x-1}{n-1}, 0\leq x \leq 1, n=1,2,3..$$ Question convengence. I do not know how to do convergence in ...
2
votes
1answer
55 views

Show that $(\frac{S_1}{S_n+1},\frac{S_2}{S_n+1},…\frac{S_n}{S_n+1})=_d (U_{(1)},U_{(2)},…,U_{(n)})$.

Let $(X_1, X_2,...,X_n) \in \mathbb R^n$ have density function $p(x)$. (1) Find the density of $(U_{(1)},U_{(2)},...,U_{(n)})$, the order statistics from a sample of iid $\mathbb U[0,1]$ (uniform ...
0
votes
1answer
33 views

Existence of the inverse of a distribution function of a real-valued random variable

In the proof of the probability integral transform at Wikipedia, they take any real-valued random variable $X$ and use the inverse of its distribution function $F_X$. However, a distribution ...
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2answers
58 views

Example of using Delta Method

Let $\hat p$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success. (a) Compute the expectation of $\hat p (1-\hat p)$. (b) Compute the approximate ...
0
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1answer
31 views

Why do these set of random variables converge in probability to $-1$?$(Y_n)$

$$Y_n=\frac{X_n}{1-X_n}=-1-\frac{1}{X_n-1}. X_n: \mathcal U(0,n)$$ So they unclear part is just the last equality in this series of equalities: $$P\{|Y_n+1|\geq \varepsilon\}=P\{\frac{1}{|X_n-1|} \...
0
votes
1answer
29 views

Convergence of this set of random variables $Y_n$ where: $X_n: \mathcal U(0,1)$ independent random variables, $Y_n=\frac{1}{nX_n} n=1,2,…$

Convergence of this set of random variables $Y_n$ where: $X_n: \mathcal U(0,1)$ independent random variables, $Y_n=\frac{1}{nX_n}\ \ n=1,2,...$ I can easily prove that this does not converge almost ...
2
votes
1answer
34 views

Why does $(W_t)^2$ have mean $t$?

This is in the context of Ito Calculus. Here, $W_t$ is a $P$-Brownian Motion. My book says that "... $(W_t)^2$ has mean $t$, because of the variance structure of Brownian motion.." I understood that ...
0
votes
1answer
44 views

Simple Question about Random Variable with Finite Mean

Consider a random variable $X$ with $E[X]< \infty$ and probability density $f_X$, I was wondering: given any positive number $\epsilon >0,$ do we always have $$ \int_{-1/\epsilon}^\infty x ...
1
vote
1answer
30 views

Distribution function of the random variable $X\cdot\mathbf 1_{X>K}$ with $K>0$

Let $X$ be a non-negative random variable and $K$ be a fixed positive number. Consider $$ Y= \begin{cases} X,& \text{if } X>K\\ 0, & \text{otherwise} \end{cases} $...
2
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0answers
50 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} \mathbb{...
4
votes
2answers
130 views

My proof that $S_n/\sqrt n$ does not converge in probability

I'm given a sequence $(X_n)$ of i.i.d. random variables with mean $0$ and finite variance $\sigma^2$. Let $S_n=X_1 + ... + X_n$. I have to show that $S_n/\sqrt n$ does not converge in probability. ...
3
votes
2answers
103 views

Origin of the notation for statistical divergence

The unusual notation $D(P||Q)$ seems to be universally used for statistical divergences (e.g. KL divergence). What is the origin of this notation, and do the double bars (pipe symbols) have any ...
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0answers
19 views

Markov-like inequality for bias of 0-1 random variables

Let $S \subseteq \{0,1\}^d$ for some integer $d$ and let $X_s$ be a random variable over $\{0,1\}$ for every $s \in S$. Define $\omega = \frac{1}{|S|}\sum_{s\in S}\text{bias}(X_s)$, where $\text{bias}(...
4
votes
1answer
93 views

Absolute Continuity of the sum of two Cantor random variables

If we have two independent random variables each having a Cantor distribution is there an easy way to see that the distribution of their sum is not absolutely continuous? I am pretty sure that if ...
1
vote
1answer
42 views

Law of total expectation well-defined?

Wikipedia states that this is a special case of the law of total expectation click me. Given a partition $A_1,...,A_n$ of the outcome space, we have for a random variable $X$ that $E(X)=\sum_{i=1}^{...
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0answers
42 views

Autocorrelation function of a Wiener process & Poisson process.

Can anyone possibly explain step 3 and 4 in this solution?
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0answers
34 views

Markov chain: Find expected value to get back to starting state

I wonder why they complicate this solution? Call the mean time to get from i to j $M_{i,j}$ and set up three simple equations starting with $$M_{0,0} = 1 + (1/3)M_{1,0} + (1/3)M_{2,0}$$ and you get ...
3
votes
2answers
41 views

Expressing equal probability on an infinite line with probability axioms

Is there any way using the usual (Kolmogorov) axioms of probability to describe/model the following situation : A value $v \in \mathbb{R}$ has an equal probability of being measured anywhere in the ...
2
votes
1answer
71 views

Conditional pdf of product of two exponential random variables

I have two independent random variables say $X$, $Y$. Both of them follow exponential distribution with parameter $λ$ i.e $X\sim λe^{−λx}$ and $Y\sim λe^{−λy}$. I want to find the pdf of $Z=XY$ given $...
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1answer
36 views

Distribution of $X\cdot Y +a\cdot X$ for $X,Y$ standardnormal

I am searching for the exact or asymptotic CDF of the rv $X\cdot Y +a\cdot X$ with the $X,Y$ independent standard normal rv's. Found nothing till now. Any hints? Thanks.
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3answers
29 views

Comparing results after performing W/ and W/O replacement on an experiment

In this is famous example in the probability theory, there are 6-Red, 4-Green, and 5-Blue balls in a bag. By calculating the probability with and without replacement for these three colors, we ...
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1answer
61 views

M/M/1 or M/M/n?

In a queuing systems with a single queue that receives $n$ poisson arrival streams with arrival rates $\lambda_1, \lambda_2, ...,\lambda_n$, and exponential service rates of $\mu_1,..., \mu_n$, we can ...
2
votes
1answer
52 views

Problem calculating MGF (expectation) using just the definition.

Assume $ (\Omega,\mathcal{F})=([0,1],[0,1]\cap\mathcal{B}(\mathbb{R})$ . Let $(X_j)_{j\geq1}$ be a sequence of independent random variables s.t. $\mathbb{P}(X_j=k)=\frac{1}{3}, k=0,1,2, j=1,2,..$ ...
0
votes
1answer
44 views

Almost sure convergence of sum of countable number of sequences

I have a countable number of infinite sequences of random variables, $(X_n^i)_{i=1}^\infty$. I know that each sequence converges almost surely: for all $i \geq 1$, $$ X_n^i \overset{\text{a.s.}}{\...
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2answers
37 views

Understating binomial and uniform distributions.

Speaking of probability distribution, Can someone kindly tell how and when I use binomial distribution and uniform distribution in real life situations? I understand their mathematical formulas but I ...
3
votes
0answers
45 views

Necessary and Sufficient Conditions for a CDF

This is an attempt to prove Theorem 1.5.3. in Casella and Berger. Note that the only things that have been proven are really basic set-theory with $\mathbb{P}$ (a probability measure) theorems (e.g., ...
0
votes
1answer
36 views

Convolution domains probability theory

Problem 1.4 here: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/quiz-2/MIT6_041SCF13_quiz02....
3
votes
1answer
68 views

Probability man women in a survey

In a survey, we asked $7$ men and $5$ women. Is randomly selected without replacement persons one by one until a man. Let $X$ be a random variable of the number of prints required. Determine ...
2
votes
0answers
50 views

Finding limiting distribution of $\sqrt n \frac{(S_n-5n)}{\sum_{i=1}^n X_i^2}$

Let $X_1, X_2,...$ be IID rv with $E[X_1] = 5$ and $Var[X_1] = 9$ Find the limiting distribution of $\sqrt{n} \frac{(X_1 + X_2 + ... + X_n - 5n)}{(X_1^2 + X_2^2 + ... + X_n^2)}$ as $n \rightarrow \...
0
votes
1answer
106 views

What is a “continuous modification”? And can we always modify an almost surely continuous process, such that every path is continuous?

Let's motivate the question by a classical result: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ which ...
1
vote
1answer
53 views

About convergence in probability

A few days ago, I was introduced the concept of convergence in probability, after the almost sure convergence. I understood the almost sure convergence (I think): We have a sequence of random ...
3
votes
1answer
79 views

Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
1
vote
1answer
53 views

Find expected value of a dice experiment

The following experiment is performed, Roll a dice. If you stick to the outcome, then the final score is the number on the dice. The experiment ends here. If the experiment is performed $n$ times, ...