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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2
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0answers
30 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
1
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1answer
27 views

Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
0
votes
1answer
23 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
1
vote
1answer
23 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
1
vote
1answer
49 views

How long would it take to a lottery number repeat?

In Professor Stewart’s Cabinet of Mathematical Curiosities the following is asked: You have $1000$ songs on your MP3 player. If it plays songs ‘at random’, how long would you expect to wait ...
0
votes
1answer
83 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
1
vote
1answer
35 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
-1
votes
0answers
42 views

Probability_distribution [closed]

Three points are chosen at random on the circumference of a circle. Find the probability that they all lie on the same semicircle, using random numbers generated from a uniform distribution.
0
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1answer
29 views

4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
0
votes
1answer
68 views

incorrect rejection of a true null hypothesis? [closed]

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
0
votes
1answer
27 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
0
votes
1answer
25 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
2
votes
0answers
33 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
0
votes
1answer
19 views

Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
0
votes
1answer
37 views

Representing the probability as a recurrence equation

Introduction Suppose that you initially have an $n$-sided die with equal probability and you throw it then you will get a certain number $1< k \leq n$ then you throw a $k$-sided die. Continue ...
1
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0answers
22 views

conditional expectation of exponential random variable conditoned on sum of exponential random variable

Let X,Y be i.i.d. exponentially distributed with parameter $\lambda$. Show that for $Z:=X+Y$ and a measurable, non-negative function $h$ we have: ...
1
vote
1answer
23 views

Local martingale being true martingale

I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale. In ...
1
vote
0answers
17 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
3
votes
1answer
44 views

Reference request for stochastic process

I studied the book, "Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal estimation and detection. In ...
0
votes
2answers
20 views

bayes theorem related problem. [closed]

Suppose that Mr. Zafar becomes sick in the middle of the night and asks his sleepy wife to get some drug for him from the medicine cabinet. Two kinds of tablets are available T1 and T2. There are only ...
14
votes
1answer
205 views
+100

Show two random variables have same distribution

Let X, Y be two non-negative random variables satisfying the condition $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. How can one show that X and Y are equal in ...
3
votes
3answers
39 views

You are making cookies and add N chips to dough randomly, and split it into 100 equal cookies, again at random. How many chips should go into dough?

Question: You are making chocolate chip cookies. You add N chips randomly to the dough and you randomly split the dough into 100 equal cookies. How many chips should go into the dough to give a ...
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votes
0answers
18 views

Proof $Var(X|Y)=E(X^2|Y)-E^2(X|Y)$ [closed]

Is it true and if yes, then how to prove it (please do not use the law of total variance because I need this relationship to prove the law itself): $$Var(X|Y)=E(X^2|Y)-E^2(X|Y)$$
3
votes
1answer
37 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
0
votes
2answers
27 views

Finding the probability of a complex event…

My apologies - I'm not a math professional, but the guys in my office (a bunch of web programmers) just came across across a logic problem that we've been discussing. We have a solution, but now ...
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votes
0answers
12 views

probability theory and application btech 2nd year mike [closed]

A class consisting of 4 graduate and 12 undergraduate students is randomly divided into 4 groups of 4. What is the probability that each group includes a graduate student?
1
vote
1answer
31 views

Probability of exactly $K$ events out of possible $N$

So I've stumbled upon this question in Grimmett and Stirzaker's text. I have their solutions manual, which starts off like this: The line above, where the statement is expanded into sums, is where ...
-2
votes
2answers
38 views

Exponential distribution of random variable [closed]

Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with ...
1
vote
1answer
32 views

Transitivity of a stochastic order

Let $X$, $Y$, $Z$ be three independent random variables such that $P(X \geq Y) \geq 1/2$, $P(Y \geq Z) \geq 1/2$. Is it true that $P(X \geq Z) \geq 1/2$? It seems true but I'm having a hard time with ...
-1
votes
2answers
60 views

Expected values of a dice game with a 30-sided die and a 20-sided die.

Two people, $A$ and $B$, have a $30$-sided and $20$-sided die, respectively. Each rolls their die, and the person with the highest roll wins. ($B$ also wins in the event of a tie.) The loser pays ...
0
votes
1answer
51 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
2
votes
3answers
433 views

Product of two infinite sequences

Let $p_i$ be reals in (0,1) such that $\sum_1^{\infty} p_i=\infty$ and $\sum_1^{\infty} (1-p_i)=\infty$. Prove that $\sum_1^{\infty} p_i(1-p_i)=\infty$. I know a probabilistic proof (follows from ...
1
vote
1answer
22 views

The hierarchy of Liapounov conditions

The general setup is an array $(X_{nj} : n \in \mathbb{N}, 1 \leq j \leq k_n)$ of random variables (where of course each $k_n$ is an integer of value at least $1$). Write $S_{n} := \sum_{j=1}^{k_n} ...
0
votes
0answers
41 views

Find probability of complex event.

Okay, so today I've had task in my test, which is supposed to be hard: Cafe serves donuts. Every day number of students eat in this cafe. $\frac{3}{5}$ of these students are engineers, and ...
2
votes
1answer
57 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
1
vote
1answer
46 views

Computer Component with Gamma Distribution? [on hold]

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...
1
vote
2answers
31 views

Two company and probability example?

I ran into a problem that seems strange to me. Two companies A,B produce a device that with probability $0.05$ and $0.01$ are broken. if we buy two devices produced by one company with equal ...
2
votes
2answers
46 views

Conditional expectation and absolute continuity

Let $(\Omega,\mathscr{F})$ be a measurable space and $P$, $Q$ be two probability measures. Assume $Q$ is absolutely continuous with respect to $P$ and $\mathrm{d}Q/\mathrm{d}P=f$. I will use $E^P$ and ...
10
votes
1answer
65 views

Given $\mathbb{E}[X|Y] = Y$ a.s. and $\mathbb{E}[Y|X] = X$ a.s. show $X = Y$ a.s.

Given $X,Y \in L^2(\Omega,\mathscr{F},\Bbb{P})$ such that $\mathbb{E}[X|Y] = Y$ a.s. $\mathbb{E}[Y|X] = X$ a.s. show that $\Bbb{P}(X = Y ) = 1.$ $Attempt: $ I can see that $\mathbb{E}[X|Y] = Y$ ...
4
votes
1answer
31 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
2
votes
1answer
38 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
0
votes
1answer
18 views

Independence of a random variable and a sub-$\sigma$-algebra

I am having trouble understanding one of the steps in the proof of the following lemma. Let $X$ be a random $d$-vector and $\mathcal{A}$ a sub-$\sigma$-algebra on the probability space ...
1
vote
1answer
35 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
0
votes
0answers
69 views
+50

Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
0
votes
1answer
17 views

Outer Measure on a Probability Space is 1 iff its complement is null?

I am trying to prove the following theorem, which I feel should be true, but am not sure how to go about it. Suppose we are in a probability space $(\Omega, \mathcal{F}, P)$ and we define for any ...
0
votes
1answer
30 views

How do I compute $P(X=Y)$? for independent random variables with with geometric distribution.

let $X$ and $Y$ be independent random variables with geometric distribution and parameter $p\in(0,1)$ How do I compute $P(X=Y)$? Any help would be greatly appreciated.
1
vote
2answers
26 views

Show $\lim\limits_{m\to\infty}P(n\leq m)=1$ for some function $n:\Omega\to\mathbb{N}$

Suppose that $(\Omega,\mathcal{F},P)$ is a probability triplet and $n:\Omega\to\mathbb{N}$ is some measurable function (in particular, $n(\omega)$ is finite for each $\omega\in\Omega$). I'm ...
0
votes
0answers
8 views

Counting minimal cut sets in nonamenable graphs

Suppose that $G$ is a fixed infinite, bounded degree, countable, connected nonamenable graph, meaning that its Cheeger constant is positive. Let $x\in G$ be a fixed vertex. I need to show (for ...
1
vote
2answers
53 views

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$?

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$? I thought that by monotonicity, which states that if $A \subseteq B$ then $P(A) \le P(B)$, then: If $P(A) < P(A \cup ...
1
vote
0answers
43 views

Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...