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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20 views

How to find mean and variance for probability problem with warranty?

I am in a probability theory class and I'm stumped on a problem: A warranty is written on a product worth \$10,000 so that the buyer is given \$8000 if it fails in the first year, \$6000 if it fails ...
1
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0answers
11 views

Proof of Wald's Identity, is this valid?

So Wald says that assuming that $T$ a stopping time and $X_i$ i.i.d. variables are $L^1$, that $E[S_{T}] = E[T]E[X_1]$ given that $S_n = \sum_{i=1}^n X_i$. Consider the following proof that is ...
3
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0answers
32 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
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0answers
9 views

Cauchy distribution derivation

So now I'm doing a different example with the Cauchy distribution by letting $Z=\tan(U)$ for $U$ distributed between $[−π/2,π/2]$. So then $P(\tan(U)≤a)=P(U≤\arctan(a))$, which is equivalent to the ...
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1answer
41 views

How to approach analyzing probability problems? (Specific question included)

I've recently become very interested by the concept of probability. After doing some studying, I believe I've become fairly familiar with the terms: probability, random variables, probability ...
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0answers
15 views

Kullback-Leiber divergence for two simple probability vectors

For any probability vectors $ p= (p_1,...,p_K) $ and $ q=(q_1,...,q_K) $ representing monotonically increasing functions $ x-1 $ and $ ln(x) $ respectively what is the KL divergence? $$ KL(p||q) = ...
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1answer
18 views

Probability density function of function of a random variable [closed]

What is the probability density of say $e^{a + bX}$ if $X$ is normally distributed?
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1answer
15 views

Doob Decomposition is $L^1$ bounded

Suppose $X_n$ is a martingale that is $L^p$ bounded for some $p > 1$. Then the problem asks to show that the Doob Decomposition of the submartingale $|X_n|^p = M_n + A_n$ where $M_n$ is a ...
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1answer
62 views

Conditional expectation w.r.t. random variable and w.r.t. $\sigma$-algebra, equivalence

Let $\Omega = \{ \omega_i \}$ be a countable set, and consider some probability space $(\omega, \mathcal F, P)$ with $p_i := P(\{ w_i \})$. Let $X : \Omega \to \mathbb R$ be a random variable, then ...
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1answer
10 views

Convergence in distribution of a normalized Poisson distributed random variables

Show using the central limit theorem that $\frac{X_n-n}{n^{1/2}}\rightarrow Z$ where $Z$ is standard normally distributed and $X_n$ is $Poisson(n)$ distributed.
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1answer
22 views

Law of large numbers and identically distributed variables

I have trouble understanding the need of the "identically distributed variables" hypothesis in laws of large number's type theorem. For example, here ...
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1answer
23 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
9
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3answers
266 views

Expected Value of R squared

Let $n$ be a fixed positive integer. Generate $n$ numbers $x_1, x_2, ..., x_n$ from the set $[0,1]$, with the probability distribution being the uniform one and the $x_i$ all being independent of each ...
0
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0answers
10 views

Two iid random variables [duplicate]

Prove that for every two independent, identically distributed real random variables $X$ and $Y$, we have \begin{align*} P(|X-Y| \le 2) \le 3P(|X-Y| \le 1). \end{align*}
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0answers
17 views

distribution of the maximum of independent poisson random variables.

Let $X_i$ $i=1,\dots,n$ be independent poisson random variables with $X_i \sim \text{Poisson}(\lambda_i)$ then we define $X = \max_i X_i$ how does $X$ distribute? Is easy to see that ...
1
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0answers
73 views

integral with respect to the point measure [closed]

We have integral $$\int_0^tf(t-u)dX(u)$$ where $X(u)$ is random point process( or at least renewal process). Also it is known that $f(t)\sim t^{-\alpha},$ $0<\alpha<1$ as $t\rightarrow \infty$. ...
1
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1answer
9 views

Meaning of sampling i.i.d rvs from (random) probability measure?

Quote from book: "Consider an arbitrary atomic probability measure $\Gamma$ on unit sphere. Let $(\sigma_{l})$ denote an i.i.d sample from $\Gamma$." I don't understand the second sentence. Does it ...
1
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1answer
11 views

How to show that increasing r.v. imply stochastic dominance?

How can one prove the following statement: If $X$ and $Y$ are random variables such that $X(\omega) \geqslant Y(\omega)$ for all $\omega$ then $\mathbb P(X>x) \geqslant \mathbb P(Y>x)$ ? I saw ...
0
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0answers
17 views

The conditions on marginals that guarantees a certain class of measures

$x$ and $y$ are $m\times n$ matrices. $a, B,C$ are $m\times 1$ matrices. $b, A$ are $n\times 1$ matrices. $$\sum a_i=1, 0\leq a_i\leq 1, \forall i$$ $$\sum b_j=1, 0\leq b_j\leq 1, \forall j$$ ...
1
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0answers
32 views

Discrete measures and discrete kernels

This question was also posted here. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
1
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1answer
29 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
0
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0answers
17 views

A two-stage experiment where the first stage has two independent outcomes

If $P(Y_1\in \cdot|X_1, X_2) = P(Y_1\in \cdot|X_1)$ and if $P(Y_2\in \cdot|X_1, X_2) = P(Y_2\in \cdot|X_2)$ and if $X_1$ and $X_2$ are independent, are $Y_1$ and $Y_2$ independent given $X_1, X_2$, ...
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1answer
67 views

Given $B \cup A = B$ and probability and set theory axioms, prove $\mathbb{P}(A) \leq \mathbb{P}(B)$.

I need to prove that $\mathbb{P}(A)$ is less than or equal to $\mathbb{P}(B)$ using only this three things: $B \cup A = B$ The three axioms of probability: a) $\mathbb{P}(A)$ is greater or equal to ...
0
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0answers
19 views

Proper definition of Kullback-Leibler divergence for densities.

Let $f$ and $g$ be densities on the same set $X$. The Kullback-Leibler divergence is expressed by this famous formula: $$ D(f||g) = \int_{X^{+}} f(x)\log\left(\frac{f(x)}{g(x)}\right)dx \textrm{,} $$ ...
0
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1answer
22 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...
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0answers
24 views

Set of all probability measures with finite support

Let $X$ be an uncountable set endowed with the discrete topology. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, and consider the subset $A$ of $\mathcal{P}(X)$ consisting ...
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1answer
42 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
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0answers
23 views

The limit of probability mass functions is also a mass function

Suppose $\{X_n\}$ are discrete random variables with respective mass function $\{p_n\}$. Prove that if there exist a function $p$ such that $\lim_{n\rightarrow \infty}\sum_{z\in R}|p_{n}(z)-p(z)|=0$, ...
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2answers
30 views

An $m$ sided dice is rolled $n$ times what is the chance of getting an average of $\frac{m+1}{2}$?

If I roll a six sided dice twice there is a $1$ in $6$ chance that the results will sum up to $7$ (giving an average of $3.5$ per dice). And if I only roll it once it is not possible to get a $3.5$. ...
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1answer
28 views

Optional Stopping Theorem for Stochastic Processes with Constant Mean (and not a Martingale)

Let $(X_n)_{n\geq 1}$ be a martingale with respect to $(Y_n)_{n\geq 1}$, i.e., the martingale condition $$ \mathbb{E}[X_n|Y_1, \ldots, Y_{n-1}] = X_{n-1} $$ holds. From this condition, it follows ...
0
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1answer
49 views

Expected value of the product of i.i.d random variables

Assume we have random variables $$X_i \,\,\,\ \text{ i.i.d } \,\,\ i\in[1:n]$$ with expected value $$\mathbb{E}[X_i] = \frac{1}{2}$$ Now let us compute the following expected value of the product of ...
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2answers
8 views

Expected delay problem on expectation based on uniform distribution

At a traffic junction, the cycle of traffic light is 2 minutes of green and 3 minutes of red. What is the expected delay in the journey, if one arrives at the junction at a random time uniformly ...
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0answers
30 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
2
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1answer
57 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
3
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1answer
32 views

Does wikipedia state the definition of probability correctly?

In the wikipedia article on probability http://en.wikipedia.org/wiki/Probability it says: To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a ...
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0answers
151 views
+50

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
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1answer
26 views

if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$?

In handouts provided by a professor I read: if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$. It does not feel right to me. $X ...
-1
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0answers
13 views

Odds of wining sweepstakes vs price of ticket

I've seen lots of similar questions but none answer my question. There is an online raffle (so no bend ticket answers) with $2000$ tickets total. Each ticket costs $5$ points, $1$ prize of $5000$ ...
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2answers
44 views

What is the meaning of the below probability equation?

Can someone explain the intuitive idea behind this probability equation (especially the part where the limit of epsilon downarrow zero notation).
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1answer
22 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
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0answers
41 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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0answers
17 views

Showing that Central Limit Theorem is valid, Lyapunov condition

Let $Z_1, Z_2, \ldots$ be independent random variables, $Z_k \sim U[-1,1], ~k=1, \ldots, n$, $(a_n)_{n \in \mathbb N}$ sequence of positive numbers, $\lim_{n \rightarrow \infty} ~ na_n = \infty$. I ...
0
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3answers
28 views

Is this a conditional probability or not?

Suppose that the telephone calls during one minute time follow a Poisson distribution with mean=4. If people can handle at most 6 calls per minute, what is the probability that the people will receive ...
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2answers
37 views

infinite sum of normal r.v. is still a normal r.v. when given $\sum \limits_{i=1}^\infty a_i^{2}$ is finite

If $X_1, X_2, ...$ are i.i.d.standard normal random variables and for real constants $a_1, a_2, ...$, given $\sum \limits_{i=1}^\infty a_i^{2} $ is finite, then $Y_n =\sum\limits_{i=1}^n a_iX_i$ ...
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0answers
34 views

Proof of the Box-Muller method

This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly ...
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1answer
35 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
0
votes
1answer
37 views

What are the possible values of Z=X+Y

If I have two independent probability mass function, where $P_{x}(0)=\frac{1}{2}$ , $P_{x}(2)=\frac{1}{2}$ and $P_{y}(1)=\frac{1}{6}$ , $P_{y}(2)=\frac{1}{3}$ , $P_{y}(3)=\frac{1}{2}$ I am asked ...
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0answers
37 views

“Expected value” of Thirteen card game !! [duplicate]

Thirteen cards numbered $1$ to $13$ are shuffled and dealt one at a time. "Match" occurs on deal $k$ if $k$th card revealed is card number $k$ Let N be the total number of matches that occur in the ...
2
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1answer
33 views

Convergence in distribution conditions

I'm reading the weak convergence section (Ch.$18$) in "Probability Essentials" by Protter and Jacod. Theorem $18.7$ states that "$Xn \overset{D}{\to} X \iff \underset{n\to\infty}{\lim} E[g(X_n)] = ...
2
votes
0answers
83 views

Almost surely either $X_n=0$ for some $n$ or $\lim_{n\to\infty}X_n=\infty$

How I came to this: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of non negative random variables, $\mathcal{F}_n=\sigma(X_l,l\leq n)$ the sequence of corresponding sigma-algebras and define ...