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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Does a state which is passed at least 3 times had to be passed 5 times in Markov chain

Prove of disprove: Let $\{X_n\}_n$ be homogenous Markov chain. if we start from state $i$, there is a positive probability that we pass at least 3 times at state $j$. Does it follows that exists ...
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1answer
18 views

Showing that a collection of intervals (see problem) generates the Borel sigma algebra on $(0,1]$

I would be very appreciative if someone could show me how to do this problem so that I can try to get a better understanding of what a Borel sigma algebra is. Examples are how I learn best so seeing ...
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1answer
20 views

Probability of two elements falling within a certain interval

I did this and said that $a$ must be in $(2, 3)$ and $b$ must be in $(-2, -1)$. Then the probability that $b$ is in that interval is $\frac{1}{2}$ and the probability that $a$ is in $(2, 3)$ is ...
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1answer
13 views

Applying Ito's formula to a complicated expression

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t ...
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1answer
18 views

How does a change of measure affect covariance?

Suppose I have the three random variables $X,Y,M$ where $E[M] = 1$ under the measure $P$. Now, suppose I define a new measure $\widetilde P$ so that $\widetilde E[X] = E[M X]$ and $\widetilde E[Y] = ...
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8 views

local martingales and dividend processes

Consider a $d+1$ asset, continuous-time model where asset $0$ is a riskless numeraire. Assume that the asset prices are modelled by a $(d+1)-$dimensional Ito process (B,S). Further, let $D$ be the ...
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13 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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10 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
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1answer
26 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
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16 views

Convergence of Uniformly Distributed Random Variables (n-dimensional)

Suppose that ${U_n} = ({U_{n1}},{U_{n2}},...,{U_{nn}})$ is uniformly distributed over the n-dimensional cube ${C_n}={[0,2]^n}$ for each $n=1,2,...$ That is, that the distribution of ${U_n}$ is ...
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4 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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0answers
20 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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1answer
20 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
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0answers
12 views

Exercise on Markov chains

I'm preparing my Probability exam and I'm having trouble with exercise 2 here. The question is to consider the random walk on $E$ with transition matrix $p$ and find the communication classes (or ...
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0answers
16 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
21 views

Probability of drawing balls from an urn with variable composition

A coin is tossed $k$ times, with probability $p$ of heads. In an urn, as many white balls are introduced as the amount of heads obtained, and as many black balls are introduced as the amount of tails ...
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2answers
58 views

Is it true that E [ X | E [ X | Y] ] = Ex [ X | Y] ? Does this law have a name?

Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true: E [ X | E [ X | Y ] ] = E [ X | Y] Note that E [ X | Y ] is a ...
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0answers
14 views

Proving criterion for a transient state in Markov Chain

Let $\{X_n\}_n$ be a homogenous Markov chain. Prove that if exist a connected subset of states (means set of states which exist positive probability to move between them), $S$ which is not closed, ...
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3answers
62 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ My current reading material claims ...
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0answers
23 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
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0answers
18 views

discrete values probabilities problem [on hold]

Can you help me with this?? An engineer is requested to design water supply and waste water removal systems in a new industrial park consisting of 5 independent buildings. Assume that the water ...
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0answers
16 views

Clarification on product space, conditional probability measures

Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ ...
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1answer
14 views

Showing that a sequence converges, in distribution, to a normal r.v.

Let $X_1, X_2, \dots$ be independent, with $$P(X_n=1) = P(X_n = -1) = \frac{1}{2} \left( 1 - \frac{1}{n^2} \right),$$ and $$P(X_n = n) = P(X_n = -n) = \frac{1}{2n^2}.$$ Let $S_n = X_1 + \cdots + ...
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1answer
42 views

A Question on CDFs and PDFs (substitution/inverse?)

(a) So there has been an answer to the question. Can someone explain how the limits of integration were found? I don't know why the upper limit is going to $X$.
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16 views

Suppose that there are two cells in a parallel system. In order for the system to work, at least one of the two parallel subsystems must work.

Consider a particular lifetime value $t_0$, and suppose we want to determine the probability that the system lifetime exceeds $t_0$. Let $A_i$ denote the event that the lifetime of cell $i$ exceeds. ...
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0answers
22 views

Finding a random variable.

Let $X,Y$ be two non negative random variables such that density of $Y$ is the same as the survival function of $X$. Is there any way we can find $Y$? Thank you for your time and help.
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1answer
43 views

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I've done: By conditional Jensen ...
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1answer
26 views

Probability Theory: Expectation Problem [on hold]

Need help. (1) Can we mathematically proof that the difference between $\mathbb E[\min(X,Y)]$ and $\min(\mathbb E[X],\mathbb E[Y])$ is minimal / negligible? (2) Can we say that $\min(\mathbb E[X], ...
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1answer
37 views

what could 1/probability represent?? [on hold]

I was working on a concept in probability theory with a friend and we came across 1/probability. Does the inverse of probability appear anywhere in mathematics and what are its applications?
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18 views

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
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1answer
8 views

Expectation and variance of X − Y

Let's say I have $X=\min\{X_1,...,X_{10}\}$ with the $X_i\sim Exp(\lambda_i)$ independent. And let $Y=\min\{X_{11},...,X_{20}\}$ What is the expectation and variance of $X-Y$? I really don't know ...
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1answer
26 views

$X_n \to X$ in $L_2$, show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$

$X,X_1,...$ are random variables, $X_n \to X$ in $L_2$. Show that $\lim_{n \to \infty}E[X_n^2]=E[X^2]$. My attempt: $X_n \to X$ in $L_2 \implies \lim_{n \to \infty} E[(X_n-X)^2]=0 \implies \lim_{n ...
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1answer
27 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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19 views
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1answer
25 views

what is the difference between event space and probability space?

Let the sample space, $S=\{1,2,3,4\}$ and event space,$F$ is defined on $S$ are $\{1\}$ and $\{2\}$.Enumerate all possible events in $F$. This is the question I encountered while solving problems ...
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9 views

Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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1answer
29 views

Given the distribution of $X$ and $Y=-2\theta \ln X$. How is $Y$ distributed?

The pdf of $X$ is $f(x) = \theta x^{\theta-1},\enspace 0<x<1, \enspace 0<\theta<\infty.$ Let $Y=-2\theta \ln X.$ How is $Y$ distributed? My work: $$ \begin{align*} F(Y) = P(Y \leq y) ...
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1answer
24 views

Please check my answers on these Convergence Problems

Let $X_1,X_2,...$ be a sequence of random variables with corresponding distribution functions given by $F_n(x)=0$ if $x<-n$, $F_n(x)=\dfrac{x+n}{2n}$ if $-n\leq x< n$ and $F_n(x)=1$ if $x\geq ...
3
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1answer
32 views

Exercises with solutions for probability theory?

I'm reading the book Probability Theory: A Comprehensive Course by Achim Klenke. There are no solutions for the exercises in this book, so I constantly have to annoy people here (but nobody wants to ...
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3answers
34 views

Expectation of min of two random variable

Looking for your kind help to solve the following expectation problem. Let assume, $C_{u} = min(C_{a},C_{b})$ where $C_{a}$ & $C_{b}$ are random variables. Is the following is true? ...
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1answer
50 views

Markov chain doesn't sum up to 1

Let $\{X_n\}$ be a Markov chain on $S=\{1,2,3,4,5,6\}$ with the matrix suppose we define a new sequence $\{Y_n\}$ by $$Y_n=\cases{1\quad X_n=1\vee X_n=2\\2\quad X_n=3\vee X_n=4\\3\quad ...
3
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0answers
24 views

Existence of regular conditional distribution of random variable given the value of another variable

Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, ...
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15 views

Have I used the Probability generating function of poisson point process correctly?

Let $v\in \mathcal{V}$ be measurable and let $\Phi$ be a Poisson Point Process with intensity $\lambda$ then the probability generating function (PGF) is $$\mathbb{E}\left( \prod_{x\in \Phi} ...
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1answer
12 views

Is the product measure space generated by the filtration adapted to the projection maps?

Let $(\Omega, \mathcal A)$ be a measure space. Consider the product measure space $(\Omega^{\mathbb N}, \mathcal A^{\mathbb N})$ and denote by $\pi_n : \Omega^{\mathbb N} \to \Omega$ the $n$-th ...
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1answer
37 views

Find the Expected value of a Random variable

Assume random variable $$X \sim f_X(x) \,\,\, -2 \leq x\leq 2$$ Now Assume we need to compute the following $$F= \mathbb{E}\left(\frac{1}{1+(G(X))^2}\right)$$ where we define the function $$G(x) = ...
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22 views

Show Projection minimizes variance

Van der Vaart's Asymptotic Statistics, problem 11.2 Another idea of projection is based on minimizing variance instead of second moment. Show that $\text{Var}[T-S]$ is minimized over a linear space ...
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21 views

Central limit theorem with Lyapunov condition

$Z_1, Z_2,...$ are iid uniformly distributed on $[-1;1]$, $\lim_{n \to \infty} a_n=0$ and $\lim_{n \to \infty} na_n=\infty$ also $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
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1answer
19 views

Probability issue given a Bayesian Network

If we have a Bayesian Network A -> B ->C then P(B|A, C) = P(B|A)? Thanks!
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14 views

Proof of theorem on markov chains

If $X=(X_n)_{n\in\mathbb N}$ is a Markov chain on a space $E$, it has an initial distribution $(\lambda_i)_{i\in E}$ such that $\sum\lambda_i=1$ and a transition matrix $(p_{ij})_{i,j\in E}$ such that ...
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2answers
50 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...