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 ...

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1answer
14 views

Confusion about calculating probability of at least one event occurring

The probability that Tom will win the Booker prize is 0.5, and the probability that John will win the Booker prize is 0.4. There is only one Booker prize to win. What is the probability that at least ...
2
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2answers
54 views

Estimating $\mathbb P\{\max_{1\le j\le n}\lvert S_j\rvert\le t\}$, so called Charles Stein's theorem?

Problem (Kai Lai Chung, A Course in Probability Theory, section 5.5, Ex6) Suppose $\{X_n\}_{n>0}$ is a sequence of i.i.d. random variables. $S_n:=X_1+\dotsb+X_n$. For each $t>0$, define ...
1
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1answer
23 views

Normal approximation of binomial distribution

Problem: On average, every 50th shell has a pearl. What is the minimum amount of shells you have to open to get at least one pearl with probability greater or equal to 0.95. Calculate using the ...
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0answers
14 views

Partial derivative of a random vector

If $x$ indicates a $1\times n$ random vector of any distribution, then is the partial derivative of $x$ w.r.t $x$ equal to the derivative of the individual elements in the matrix, or are they just the ...
3
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1answer
20 views

Martingales and different definitions

Are there any differences between the following definitions of Martingales and if so what are they? Let $(X_{i})_{i=1}^{n}$ and $(Z_{i})_{i=1}^{n}$ be sequences of random variables then $(X_{i})$ ...
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1answer
79 views

Find the characteristic function of combination of random variables

I have the same problem as here: Find characteristic function of random variable. Could you explain last equality? Can I get it without using law of total expectation? Update I have some idea, $E $$ ...
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0answers
23 views

Cramer-Rao-Bound for squared parameter

I have a random sample $X_{i}, \ldots, X_{n} \sim \mathcal{N}(\theta,1)$. I would like to find the Cramer-Rao lower bound for the variance of unbiased estimators of $\theta^{2}$. Here's what I have: ...
2
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0answers
36 views

Expected magnitude of a vector of $n$ i.i.d. random variables as $n\to\infty$

Suppose that $X_i$ are i.i.d. real valued random variables with probability distribution $f(x)$ for $i=1,2,3,\ldots$. Let $Y_n=\left(\sum_{i=1}^nX_i^2\right)^{1/2}$. Assuming that ...
2
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1answer
70 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...
0
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1answer
18 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
2
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0answers
46 views

Rigorously, what is the goal of (machine/statistical) Learning and why is that the goal?

After some time doing machine learning and statistical learning theory, I decided to return to my foundations and make sure that the goal of what I am doing makes sense. First let me define $I(f)$ as ...
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1answer
34 views

Is square of Wiener process an orthogonal process?

I'm trying to prove: Let $t_1 < t_2 \leq t_3 < t_4$ and $(X)_t$ is the square of Wiener process. Then $E(X_{t_2} - X_{t_1})(X_{t_4}- X_{t_3}) \neq 0.$ Progress Maybe the fact $E(X_{t_2} - ...
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0answers
17 views

Sampling from general multivariate Gaussians

Suppose you have access to a sampler such as randn in Matlab/Octave that returns samples from a simple one-dimensional Gaussian distribution (a normal distribution) ...
2
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0answers
33 views

Questions regarding Martingales

I'm trying to learn about Martingales with specific focus on combinatorial problems. However i'm far from an expert in algebra and am having some trouble understanding the basic idea. I will write the ...
3
votes
1answer
31 views

Existence of a probability space

Let us assume that we are given a family of Markov chains $(X^\alpha_t)_{t\geq0}$ in continuous time. Kolmogorov's result ensures that for each $\alpha \in I$ there exists a probability space ...
1
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1answer
21 views

Convergence in distribution with exponential limit distribution

Let $X_1,X_2, \ldots$ be independent, identically distributed, positive random variables with probability density function $f$, which is continuous in $(0, \infty)$ and $\lambda :=\lim_{x \searrow 0} ...
1
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1answer
35 views

An inequality involving $\mathbb{E}[|X|],\,\mathbb{E}[X^2],\,\mathbb{E}[X^4]$

My question is an exercise which appears in a book on probability theory. Thank for helping. Let $X$ be a random variable with $\mathbb{E}(X^2)=1$, $0<\mathbb{E}(X^4)<+\infty$. Prove that: ...
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1answer
31 views

Elementary question on probability

Villages A,B,C, and D are connected by overhead telephone lines joining AB, AC, BC, BD, and CD. As a result of severe gales, there is a probability p(the same for each link), that any particular link ...
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2answers
34 views

Law of large numbers for Brownian Motion (Direct proof using L2-convergence)

In “Brownian Motion” by Schilling and Partzsc, they give a HINT to prove the Law of Large Numbers for Brownian Motion (not in their solutions, fyi) by (1) Noting that ...
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2answers
27 views

Value of constant k which makes the function $f(x)=\frac{k|x|}{(1+|x|)^4}$ a p.d.f.

Let $f(x)=\dfrac{k|x|}{(1+|x|)^4}$, $-\infty<x<\infty$. Then, what is the value for which f(x) is a probability density function ? f(x) will be a p.d.f. if $\displaystyle ...
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1answer
51 views

$\limsup$ and $\liminf $ of $\sum_{k=1}^n \frac{X_k}{\sigma \sqrt{n}} $

Suppose $(\Omega, \mathfrak{F}, p)$ is a probability space; $X_n$ are i.i.d. random variables defined on $\Omega$, with $E(X_i)=0$ and $Var(X_i)= \sigma$ for all $i$. Then $$ \limsup_n \sum_{k=1}^n ...
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0answers
23 views

Is expectation also a distribution?

I'm confused about Expectation in probability theory. Is it also considered a probability law? if so, why?
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3answers
207 views

what is difference between probability and probability space?

I am a beginner in probability and started reading the relative material. I encountered the exercise question "find the probability space for tossing a fair coin till the first head is observed". so, ...
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1answer
51 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
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0answers
22 views

I have questionin from the stochastic differential equation merton model

in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$ where $\alpha$ is the instantaneous expected return on the stock; ...
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2answers
45 views

How to add two random variables?

Given that $$\begin{array}{cccc} \text{X} & -1 & 1 & 3 \\ \text{p} & 0.2 & ? & 0.3 \\ \end{array}$$ and $$\begin{array}{cccc} \text{Y} & 1 & 2 & 3 \\ \text{p} ...
1
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1answer
28 views

Variance of independent variables

Let $X_k$ and $Y_k$ be two stochastic variables whose joint distribution is the regular normal distribution on $(\mathbb{R}_2,\mathbb{B}_2)$ with mean 0 and variance matrix $\begin{align*} ...
3
votes
0answers
40 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
1
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1answer
27 views

Fatou, Dominated Convergence, etc. for nets (in relation to stochastic processes)

In textbooks on Stochastic Processes, they always seem to assume that Fatou and DCT etc. can be applied to continuous-time stochastic processes $(X_{t})_{t\in\mathbb{R}_{+}}$. But in every book on ...
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1answer
20 views

Find the probability for … [duplicate]

Suppose we uniformly and randomly select permutations from the 20! Permutations of 1, 2, 3,..., 20. What is the probability that 2 appears at an earlier position than any other even number in ...
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0answers
48 views

Relation between a.s. and L_{2} convergence

I'm working through a proof, where I need to establish that $X_{t}\overset{a.s.}{\longrightarrow}0$. All I know is that $\left|X_{t}\right|\leq\left|Y_{n}\right|+\left|Z_{n}\right|$ for $t\in[n,n+1)$, ...
2
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1answer
33 views

$L^{2}$ -limit of expression involving Brownian Motion

Let $(B_{t})_{t\geq0}$ be a Brownian Motion. I would like to prove that $\max_{n\leq s\leq n+1}\left|\frac{B_{s}-B_{n+1}}{n}\right|=\frac{1}{n}\max_{n\leq s\leq n+1}\left|B_{s}-B_{n+1}\right|$ ...
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1answer
20 views

Meaning of a probability distribution being dominated by a measure

The following comes from Ghosh & Ramamoorthi (2003) Bayesian Nonparametrics. In terms of notations, $\Theta$ is a parameter space with Borel $\sigma$-algebra $\mathcal B(\Theta)$. For ...
1
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1answer
36 views

The value of $P\bigl(|X-Y|>6\bigr)$

Let $(X,Y)$ be two dimensional random variable such that $E(X)=E(Y)=3$ & $var(X)=var(Y)=1$ & $cov(X,Y)=\dfrac{1}{2}$. Then , $P(|X-Y|>6)$ is : (a) less than $\dfrac{1}{6}$ (b) equal to ...
3
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0answers
55 views

quadratic SDE solution

I have this SDE $dX_t=[a+bX_t+sX_t(1-X_t)]dt+\frac{1}{2}X_t(1-X_t)dW_t, \, X_0=0,$ where $a,b \in(0,1)$ and let's say that $s$ is a real constant (it's actually a function of $X$, but I think I can ...
5
votes
2answers
80 views

Independence of events depend upon underlying probability model?

Consider a sample space of two coin tosses = {HH, HT, TH, TT}. Suppose that the coin is fair and therefore every outcome has probability 1/4. Now, consider another probability model where the coin ...
0
votes
1answer
37 views

Probability of $P\bigl(X+Y<\frac{1}{2}\bigr)$.

Let $X$ & $Y$ be two continuous random variables with the joint probability density $$f(x,y)=2 ,0<x+y<1,x>0,y>0$$ $$f(x,y)=0,elsewhere$$ Find the value of ...
0
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1answer
26 views

If the characteristics function of a random variable is differentiable even times then it has finite moment of even order

If the characteristics function of a random variable is differentiable $2n$ times then it has finite moment up to even order $2n$. We know the converse is correct, but how can we prove this statement? ...
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0answers
38 views

Donsker's Invariance Principle and Gambler's Ruin

Let $(S_{n})_{n\geq0}$ be a Random Walk (i.e. $S_{n}:=X_{1}+\cdots+X_{n}$, where $\mathbb{P}(X_{i}=1)=\mathbb{P}(X_{i}=-1)=1/2$). Define interpolated random walks $(S^{n}(t))_{t\in\left[0,1\right]}$ ...
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2answers
47 views

Show that $V=\frac{Z_1}{\sqrt{(Z^2_1 + Z^2_2)/2}}$ has pdf $f(v) = 1 / (\pi \sqrt{2-v^2}),-\sqrt2<v<\sqrt2$

Let $Z_1, Z_2$ have independent standard normal distributions, $N(0,1)$. If the random variable in the numerator did not also appear in the denominator this would be a t distribution. Should ...
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1answer
13 views

Examples of transient and recurrent simple random walks on trees

This is a followup to Recurrence or transience of the 1-3 tree in which I discovered that my original guess of an example for some exercises was wrong. (Those exercises can be found in ...
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2answers
54 views

Brownian Motion and Continuity

Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} ...
2
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1answer
19 views

Recurrence or transience of the 1-3 tree

The 1-3 tree is a rooted tree with only the root at level n=1, and from thereafter, $2^n$ vertices at each tier of distance from the root. However, they are not connected as in the binary tree. Put ...
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0answers
19 views

Nakagami Random variable with shape parameter $m=\infty$.

A Nakagami fading distribution $X$ with parameter $m$ is given by the following $$ X\sim f(x;\,m,1) = \frac{2m^m}{\Gamma(m)}x^{2m-1}\exp\left(-mx^2\right)$$ Then the function $S:=|X|^2$ is Gamma ...
3
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0answers
49 views

Characterize the limit of an O-U process: $dX_t = -\tfrac{\mu}{\theta} X_t dt + \tfrac{\sigma}{\theta^{1/2}} dW_t$ as $\lim_{\theta \to 0}$.

Standard O-U Formulas: Take the Ornstein–Uhlenbeck process defined by the SDE $$ dX_t = -\frac{\mu}{\theta} X_t dt + \frac{\sigma}{\theta^{1/2}} dW_t $$ where $\mu > 0, \theta > 0, $ and ...
4
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1answer
165 views

Lower bound on probability of sum of random variables

Suppose that the random variables $X_i, i = 1,2,\ldots,n$ are i.i.d. Suppose $0 \leq X_i \leq 4n^2$ with probability 1 for all $i$. Suppose that $\mathbb{E}(X_i) \geq n$ for all $i$. Show that ...
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2answers
69 views

Rigorous proof of the recursion method to compute expectations in probability.

When solving expectation problems in probability, people sometime use recursive argument, but I have never seen a proof that this argument always works. For example, what is the expected number of ...
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2answers
44 views

while finding PDF of $W=X+Y$ from given Joint PDF $f_{X,Y}(x,y)$ How to find the limits of integral?

RV $X$ and $Y$ have joint PDF: $$f_{X,Y}(x,y)= \begin{cases} 8xy & 0\le y \le x \le1 \\ 0 & \text{otherwise} \end{cases}$$ Find PDF of W=X+Y I know that I need to use : ...
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1answer
40 views

Weak Convergence If and Only If (Pointwise) Convergence of Characteristic Function

This is actually a theorem from lecture notes, with the corresponding proof. Unfortunately, it doesn't prove the last bit, or mention it at all (!), and I have a question about the penultimate bit. ...
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0answers
24 views

Regarding the expectation of a function of two random variables…

I'm trying to prove the following: If $X,Y$ have discrete p.m.f $p(x,y)$, then $\forall$ real-valued function $g$, $$E[g(X,Y)]=\sum_{x}\sum_{y}g(x,y)p(x,y))$$ I wasn't sure if my argument was ...