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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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Conditional probability of events not arising from a well-defined function on S

Given a probability space $(S,\mathcal{F},P)$ where $S$ is countably infinite, $\mathcal{F}$ is the full $\sigma$-algebra, and $\forall a \in S, P(a) > 0$, is it reasonable to speak of $P(Y|X)$ ...
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32 views

Why does $P(E) < P(F)$ imply that $E \subseteq F$?

Why does $P(E) < P(F)$ mean that $E \subseteq F$ ? My reasoning (using Venn diagrams): It is seen clearly in the below picture that even if $P(E)<P(F)$, there is still some region in E that is ...
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I am having trouble figuring out how many lambda's (births) there are in a given birth-death Markov process problem.

These questions are not for assignment. I am just confused as to how to set up the problem. I also do not need help calculating the problems at hand. I understand that in a birth and death problem, $...
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1answer
15 views

How to change the order of the limit and the expectation?

Does anyone know how to prove $\lim E[X(n)]=E[\lim X(n)]$??? Here I need to prove $\lim E[X(n)]\le E[\lim X(n)]$ and $\lim E[X(n)]\ge E[\lim X(n)]$. Based on "Fatou Lemma", I can get that $E[\...
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Gluing together 2-dimensional martingale measures to create n-dimensional martingale measure, Strassen's Theorem

Strassen's theorem states that a necessary and sufficient condition for existence of a discrete-time martingale with a finite number $n$ of given marginals $\mu_1,\ldots,\mu_n$ is that the marginals ...
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0answers
20 views

Random Variable expression with the distribution function

In my textbook, it is said that if we chose the probability space ( (0,1) , B(0,1) , P) where P is the Lebesgue measure, and we have the distribution function FX(x) of a random variable X we can find ...
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can you help me understand the inputs for this bayes theorem calculator for my scenario?

I'm trying to calculate the probability of delivering a website based on the conditional delivery of an underlying database. I entered some inputs into this online Bayes Theorem calculator: https://...
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Strong Markov property, Brownian motion

I have a question about the strong Markov property of Brownian motions. Let $(\{X_t\}_{t \ge 0}, P_x)$ be a $d$-dimensional Brownian motion starting from $x \in \mathbb{R}^d$. Let $\tau=\inf\{t>0 ...
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1answer
27 views

Probability of having drawn ace of spades after drawing a card and putting it back in the deck

I have been wondering if the probability of drawing a card from a 52 card deck, obviously 1/52 probability, is different from the following. If you get to draw a card from a 52 card deck and right ...
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A random variable formed by two Normal random variables, under a conditioning process.

Imagine two independent random variables, $X$ $\sim$ $N$ $(\mu_1$,$\sigma_1^2)$ and $Y$ $\sim$ $N$ $(\mu_2$,$\sigma_2^2)$. Now imagine a process whereby one observation of $X$ and one observation of $...
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Change of variable to calculate expected value

$X$ and $W$ are independent random variables. $$ Z=X+W $$ $$ W \sim \mathcal{N}(0,\sigma) $$ $$ E[X]=\bar{x} $$ I want to calculate $E[Z]$ with respect to the joint pdf $p(z,x)$ $$ E[Z]=\int\int (x+w)...
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34 views

Probability/ eggs in boxes

I have n boxes and n eggs, which were hidden randomly in these boxes. The probabilty, that no egg is in the right box, is $$ \sum_{k=2}^n (-1)^k \frac{1}{k!} $$ How can I get to this formula?
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1answer
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Expectation of sum is less than the second moment

Given $E[f^2(X)] < \infty$ and $X_i \sim_{iid} X$, need to show $$ E\left[ \frac{1}{n} \left( \sum_{i=1}^{n} (f(X_i) - E[f(X)] \right)^2 \right] \leq E[f^2(X)]. $$ My try: $$E\left[ \frac{1}{n} \...
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1answer
25 views

How to understand this probability equation?

$\mathbf { x } ( t ) = g ( \mathbf { s } ( t ) ; \xi ) + \mathbf { n } ( t )$, where n(t) denotes the noise or modeling error and ξ the parameters of mapping $g$ How to understand the following ...
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1answer
22 views

Do finite dimensional distributions determine the law of a stochastic process?

For $i=1,2$, let $\{X_t^i\}_{t\geq 0}$ be $\Bbb R^d$-valued stochastic processes adapted to $\{\mathscr F^i_t\}_{t\geq 0}$ on the probability space $(\Omega^i,\mathscr F^i,\Bbb P^i)$. Suppose the two ...
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25 views

is this the correct probability equation?

Event A has a probability of 80% Event B has a probability of 90% Event B depends on Event A Solution: (.80 * .90) * .80 = 0.576 probability UPDATE Here are some more details for those who ...
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1answer
19 views

how to calculate cumulative probability for *dependent* events?

The formula for cumulative probability for independent events is easy enough. Just multiply the probability of the events together. For example, 2 independent events, each with a probability of 0....
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1answer
22 views

Example of two non-mutually exclusive events that are dependent?

(1)If two events are independent that implies that they are "non-mutually exclusive". Then by using logic transposition, Non "non-mutually exclusive" events implies they are "not independent" That'...
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Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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1answer
17 views

Expectation property of martingales

Suppose $(Y_n)_{n=0}^\infty$ is a martingale of discrete random variables. Put $A:= \{(Y_0, \dots, Y_{n-1}) = (y_0, \dots, y_{n-1})\}$ In a proof I'm reading, it is claimed that $$\mathbb{E}[...
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0answers
22 views

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
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3answers
25 views

Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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Wasserstein distance between centered Gaussian mixtures

We use $\mathcal{W}_2(\cdot, \cdot)$ to denote the quadratic Wasserstien distance as defined here. Now, let $X,Y = \mathcal{N}(0,1)$ be two standard normal random variables and for $ a \in[0,1]$ let $...
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0answers
18 views

Solving a Stratonovich SDE

I am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\...
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2answers
38 views

Do these two equalities hold?

$$\mathbb P (x+y > N) = \mathbb P \left(x > \frac N2 \right) + \mathbb P \left(y > \frac N2 \right) \tag{1}$$ $$\mathbb P (\mid x+y-z \mid> N) = \mathbb P \left(\mid x-z \mid > \frac ...
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0answers
20 views

Division of two independent uniformly random variable [duplicate]

Given two independent random variable X and Y which both have uniform distribution over[0,1] I want to calculate PDF of $Z =\frac{X}{Y}$ and here is my solution: $\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)...
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27 views

A question regarding inequality between weighted averages

Suppose you have three probability density functions $p_{1}(x)$, $p_{2}(x)$, and $p_{3}(x)$. Suppose further that the expectation values of $p_{1}(x)$ and $p_{2}(x)$ are unequal $\int_{0}^{1}p_{1}(x) ...
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1answer
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What is the probability that in a group of n people chosen at random, there are at least two born in the same month of the year?

So I'm working on a probability problem: In Exercise 19 assume it is equally likely that a person is born in any given month of the year. b) What is the probability that in a group of $n$ ...
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1answer
44 views

What is the probability that the monkey will type the phrase “Call me Ishmael”?

Random events are independent events. Consider a typical computer keyboard with 82 keys. And a monkey typing on this keyboard, at random. The output of the typing would look like: jw9:.2wb0288q 1nej@...
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2answers
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Variant of the Strong Law of Large Numbers

Let $X_1,X_2,\ldots$ be a i.i.d. sequence of random variables with uniform distribution on $[0,1]$, with $X_n: \Omega \to \mathbf{R}$ for each $n$. Question. Is it true that $$ \mathrm{Pr}\left(\...
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2answers
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Special case of Bertrand Paradox or just a mistake?

I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
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0answers
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Uncorrelatedness for random elements of finitely generated groups?

Suppose $G$ is a finitely generated group, $A$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $Cay(G, A)$ on $G$ as $d$. Suppose $\{X_i\}_{n = 0}^\infty$ is a ...
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When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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1answer
16 views

A relationship between Poisson distribution and gamma distribution

We define $N(t)$ to be number of events in the interval $[0,t]$. We assume that $N(t) \sim P(\lambda t)$ for $\lambda > 0$. Let $X$ be the waiting time until the $n$-th event, we need to prove that ...
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Any criterion to determine if a probability distribution over vectors is induced by a distribution over orthogonal bases?

Let $\mathcal{O}$ be the set of all orthogonal bases of the vector space $\mathbb{R}^d$. That is, every element in $\mathcal{O}$ is a single orthogonal basis of $\mathbb{R}^d$. To make this more ...
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4answers
862 views

Drawing without replacement: why is the order of draw irrelevant?

I am trying to wrap my head around this problem: Daniel randomly chooses balls from the group of $6$ red and $4$ green. What is the probability that he picks $2$ red and $3$ green if balls are ...
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1answer
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Rigorous construction of the pointwise limit of a sequence of random variables

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let $$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$ be a sequence of random variables. Moreover, let there be an event $A \...
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Distribution of life time of a serial circuit with bulbs

Assume that we have a serial circuit with three bulbs. Each bulb's life time is exponentially distributed: $$f_{bulb}(t) =\left\{ \begin{aligned} &\lambda e^{-\lambda t} & t \ge 0\\ &0 &...
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1answer
19 views

Asymptotic distribution of median estimator when density doesn't exist

We know that when density (say $f$) exists at the median(say $\theta$) then the median estimator(say $\hat{\theta_n}$) has the following property: $$ \sqrt n(\hat{\theta_n}-\theta) \to^d N(0,1/\{4f(\...
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1answer
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Questions regarding mutual independence of events [on hold]

Have a few questions regarding mutual independence: If I have a set of events $A_1, A_2, …A_n$ that are all pairwise independent, it is possible that the events may not be mutually independent? If I ...
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Find the P.M.F. of $X_{1}+X_{2}$ given $(X_1,X_2,X_3,X_4) \sim \text{Mult}(n,4,p_1,p_2,p_3,p_4)$

I can find the marginal P.M.F.s of $X_1$ and $X_2$ but then I am lost on how to convolute the two PMF's into one PMF. I should be using convolution formulas right? because that is the only way I can ...
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15 views

For a random variable Y with PDF defined as f(y)=(kθ^k)/y^(k+1) ,y≥ θ where θ>0 and k>2 Show that f(y) is a valid pdf

I have come this far: if k>2 and θ >0 y>=θ, it is obvious that (kθ^k)/y^(k+1)>=0 Then integrated (kθ^k)/y^(k+1) (kθ^k)/y^(k+1) from 0 to infinity and got (0k)^k/ky^k if (0k)^k/ky^k = 1 then 0 = y^...
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1answer
20 views

Transforming sum of exponential variables to chi-squared distribution

Assume $X_i$ are generated with the following distribution: $$ f(x; \theta, c) = \theta^{-c}cx^{c-1}e^{-(x/\theta)^c}$$ $\theta>0$ and $c>0$ is known. Further, assume $T(X)=\sum^{n}_{i=1} X_i^...
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1answer
62 views

What kind of numbers are inside a generating open interval of the Borel $\sigma$-algebra? [on hold]

If it is enough to have all open intervals (a,b) with end points $a$ and $b$ belonging to the rational numbers, a < b, in order to generate a Borel $\sigma$-algebra on $\mathbb{R}$. Asked here: ...
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1answer
42 views

Prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$ [duplicate]

I have no idea how to proceed with proving this. If $X$ is a continuous random variable, $P(X > 0) = 1$, $E(X)$ is defined and $F(x)$ is the CDF, then prove $\lim_{x\to\infty} x . [1 - F(x)] = 0$
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1answer
28 views

Why does $ \mathbb{P}\left(X < -z\right) = \alpha \Rightarrow -z = \chi^2_{1 - \alpha}(2n) $ hold?

Assume $X_i$ are generated by $\Gamma(\theta_0,n)$ distribution, and $S_n = \sum X_i$. Further, it is known that $2 \theta_0 S_n$ follows a $\chi^2(2n)$ distribution, $\theta_0$ is known, $\theta_1 &...
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0answers
32 views

Reverse engineering distributions

Suppose I am given a measurable function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the ...
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21 views

weak convergence and CDF

we know that if $(X_n)_n$ is a sequence of real random variables, then it converges in distribution to a random variable $X$ if and only if $\lim_nF_{X_n}(x)=F(x)$ at every point $x \in \mathbb{R}$ ...
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1answer
43 views

Generate spanning tree by adding random edges

The problem is that we have $M$ nodes and at each time, we add a edge between $i$ and $j$, both of which are uniformly randomly chosen. I wonder what is the probability that there exist a spanning ...
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1answer
38 views

Markov Chain upper bound on the probability of hitting time

I encountered the following problem. $\{x_t\}$: Markov chain in discrete time; $\Omega$: a finite state space s.t. $|\Omega|=n<\infty$; $\tau_w\equiv\min\{t\ge 0\,|\,x_t=w\}$, $w\in\Omega$ (first ...