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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324 views

How can I derive the PDF from conditional probabilities?

I have some function $P(i)$ which is the probability of success for an experiment on the $i$th trial. The probability mass function for the first successful trial is: $$PMF(n) = \left( ...
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1answer
114 views

PRobability Markov chain, system of equations

I'm looking for techniques or tricks to solve a system of linear equations you get where you want to find the limiting probabilities. The system is this: $\pi_0 = 0.7\pi_0 + 0.2\pi_1 + 0.1\pi_2$ ( ...
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0answers
103 views

Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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1answer
35 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
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2answers
55 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
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0answers
52 views

PDE-Based Triangle Inequality for Optimal Transportation

Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and ...
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1answer
48 views

Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
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1answer
54 views

Random variables independent

We said that two random variables $X,Y$ are independent iff we have that for $Z = X+Y$: $$P_Z(B)=\int_{\mathbb{R}}P_X(B-s)dP_Y(s) = \int_{\mathbb{R}}P_Y(B-s)dP_X(s).$$ But I still don't get this ...
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0answers
67 views

Question on expectation and mean value

For a positive random variable we define $\mathbb{E}\{X\} = \sup\{\mathbb{E}\{Y\}: 0 \le Y \le X\}$, where $Y$ is a simple random variable (and we have already defined expectations for them) Then ...
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1answer
45 views

Limit of Poisson Distribution

Just for fun, I'm looking at the concentration of the Poisson Distribution near it's mean. For $\lambda=10$, there is a 36% probability of being within 10% of the mean. For $\lambda=100$, that ...
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1answer
32 views

Independence of variables implies no functional representation?

My question's pretty simple, I just thought the title phrases it pretty well.. Anyway, the Doob-Dynkin lemma says that $X$ is $\sigma (Y)$-measurable iff there's a measurable $f$ such that $X=f(Y)$ ...
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0answers
130 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
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2answers
474 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
2
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0answers
90 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
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1answer
63 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
2
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2answers
93 views

Distribution of suits in a 13 card hand

Let's say you have 13 cards distributed from a standard deck, find the probability of this distribution of suits: 4, 4, 3, 2, (for instance 4 hearts, 4 clubs, 3 diamonds, 2 spades). My answer was: ...
2
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5answers
276 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
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1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
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1answer
100 views

Properties of Sigma Algebras of Information up to a stopping time

first of all i want to ask whether given any two $\{\mathcal{F}_t\}$-stopping times $\sigma, \tau$ is the following properties true: (i) $\mathcal{F}_{\sigma \wedge \tau} = \mathcal{F}_{\sigma} \cap ...
3
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0answers
110 views

determine type of probability distribution

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
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1answer
35 views

Probabilistic solution Poisson problem

Let us consider the Poisson problem \begin{cases} \frac{1}{2}u''=-f\qquad\text{in}\,\,(a,b)\\u(a)=u(b)=0 \end{cases} where $f:(a,b)\to\mathbb{R}$ is continuous and bounded. We have obtained ...
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0answers
81 views

L2 norm and the Kullback distance

Let $P$ and $Q$ be two probability measures with densities $p$ and $q$ with respect to the Lebesgue measure on [0,1] such that: $0<a\leq p(x)\leq b$, $0<a\leq q(x)\leq b$ $\forall x\in $[0,1] ...
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1answer
63 views

What is the distribution of the service-starting time lag w.r.t. two concurrent customers from two parallel $M/M/1/1$ queues?

Consider two parallel, independent $M/M/1/1$ queues (denoted $Q_i, Q_j$) with identical arrival rate $\lambda$ and service rate $\mu$, using FCFS (First Come First Served) discipline. Note that the ...
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2answers
54 views

Find vector of expected values ​​and covariance matrix

For vector (X,Y) with density $f(x,y)=C exp \{ -4x^2-6xy-9y^2 \}$ find constans C, vector of expected values ​​and covariance matrix. How to do this kind of exercises?
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2answers
121 views

Maximum Likelihood Estimator for Uniform Distribution

Can somebody please explain this example to me. I am struggling to see why the likelihood is $\frac{1}{\theta^n}$ only if theta is greater than the maximum x. Furthermore, why is it the case that ...
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1answer
79 views

Product measure with a Dirac delta marginal

Let $(S,\mathcal F)$ be a measurable space, and let $\nu \in\mathcal P(S,\mathcal F)$ be a probability measure on $(S,\mathcal F)$. Fix some $x\in S$ and consider Dirac measure $\delta_x$. Would like ...
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1answer
260 views

Cauchy-Schwarz inequality with Expectations

Cauchy-Schwarz inequality has been applied to various subjects such as probability theory. I wonder how to prove the following version of the Cauchy-Schwarz inequality for random variables: ...
3
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1answer
290 views

Finding moment generating functions for a dice roll

I'm learning about generating functions and moment generating functions and I'm incredibly confused about how to actually implement the taylor-series based definition. I think perhaps practicing an ...
2
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1answer
149 views

probability theory proof of exponential chebyshev inequality

This is a question about my homework. I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF? As well as the ...
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0answers
41 views

probability theory proof of inequality

proof $Var(X + Y) ≥ 1/2 Var(X) − Var(Y)$ if $X$ and $Y$ are random variable with finite second moment. I believe it has something to do with Markov inequality or Chebyshev inequality, but I don't ...
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0answers
32 views

Conditional probability using cards

I asked a similar question earlier and the answer was helpful but I'm stuck again. You draw a card from a deck of cards. What's the probability that it is a heart given that it's red? Obviously it's ...
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1answer
31 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
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1answer
100 views

Wasserstein distance from a Dirac measure

http://en.wikipedia.org/wiki/Wasserstein_metric I would like to prove that $$W^1(μ,δx_0)=∫d(x_0,y) μ(dy)$$ let $$γ∈Γ(μ,δx_0)$$ Can we say that it is the product of its marginal distributions ...
6
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1answer
314 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
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1answer
26 views

Is a continuum of mixtures of stable distributions (e.g. normals) stable?

Take some random variable $X$ and indices $i \in [0,1]$. Let $X$ be a stable distribution, (i.e. for any copies $i,i'$, $a,b>0$, $a X_i + b X_{i'} \sim c X + d$ for some $c$ and $d$). This ...
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1answer
30 views

Random variables: functions or equivalence classes

Suppose, we have a continuous parameter stochastic process. Should I consider each of the random variables as functions or equivalence class (in a.e. sense) ?
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0answers
53 views

Warren's proof for Benford's Law

Warren has a little proof of Benford's law in Hacker's Delight. To quote: Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with ...
0
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1answer
58 views

Puzzle for Applying the Definition to a t distribution

The coeffcient of variation (CV) for a sample of values $Y_1,\ldots, Y_n$ is defined by $$ CV = S/ \bar{Y}.$$ Let $Y_1,\ldots, Y_n$ be a random sample of size $10$ from a normal distribution with mean ...
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0answers
39 views

From joint to conditional probability distribution?

I found the following expression in a paper but I'm not quite sure where does it come from: let $Y$ and $V$ be two random variables defined on the probability space $(\Omega, \mathcal{F}, ...
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1answer
54 views

Does a bijection between two sets $A$ and $B$ implies $P(a \in C ) = P(b \in C)$, if $A,B \subset C$?

I'm just thinking about it. For example, a bijection between $\mathbb{Z^*_+}$ and $\mathbb{Q}$ implies that the probability of a random real number being rational or positive integer is the same (in ...
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1answer
202 views

Find the bias for the Maximum-likelihood estimator

Let $X_1,...,X_n$ be a random sample from the pdf $$f(x|\theta) = \theta x^{\theta-1} , 0 \leq x \leq 1, \theta >0.$$ I found the Maximum-likelihood estimator of $\theta$ is $$\hat{\theta} = ...
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1answer
126 views

Markov chain notation

In a book of stochastic approximation, in the convergence of the ODE method chapter I see the following notation : the state vector of a system $X_n$ has a dynamic representation controlled by ...
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2answers
214 views

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
1
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2answers
263 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
2
votes
1answer
333 views

Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the ...
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1answer
63 views

1D biased random walk - is the event of infinte many returns a tail event?

I am considering a biased random walk: $X_1,X_2,\dots$ iid with $\mathbb{P}(X_1=1)=p$ and $\mathbb{P}(X_1=-1)=1-p$ with $p\in[0,1]\backslash\{1/2\}$, $S_n=X_1+\dots+X_n$. In this setting I want to ...
4
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2answers
81 views

If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?

Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$? What I actually want is that $X$ and ...
2
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1answer
128 views

Expectation of $x^4$ [closed]

Can anyone help me prove that Expected Value of $X^4$ is $3\,($Var$(X))^4$, if the Expected Value of $X$ is zero and Var$(X)$ is the Variance of $X$ $(N(0,\sigma^2))$.
2
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1answer
48 views

find random variables with covariance $>0$ and $=0$

I am currently studying for an exam and found the following question: Let $(x)_ {n\in\mathbb N}$ be a sequence of random variables. Can you construct a sequence of random variables such that for the ...
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1answer
49 views

Drawing without replacement yields identically distributed sequences

This question is inspired by my interest in this answer by Andre, and is related to advancing my background in combinatorics overall. How can we show formally the following fact. If we draw ...