Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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34 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
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0answers
60 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
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1answer
27 views

Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx $$ However on one ...
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19 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
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13 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
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2answers
29 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
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2answers
30 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
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0answers
38 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
2
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1answer
37 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
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0answers
44 views

How to obtain Black-Scholes from displaced diffusion process?

The displaced diffusion process is $$ d(F_t+a)=\sigma(F_t+a)dW_t $$ I have solved it and found it to be $$ F_t={F_0\over\beta} \exp\left( -\frac 12\beta^2\sigma^2t + \beta\sigma W_t \right) ...
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21 views

Usual linear combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
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1answer
29 views

Reverse Fatou's lemma on probability space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets. Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq ...
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1answer
29 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
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1answer
242 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
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1answer
95 views
+50

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
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0answers
10 views

Extensions of universal measures

Let $(\Omega,\mathcal F)$ be a measurable space, and let $\mathcal P$ be the set of all probability measures no this space. Let $\mathcal F^p$ denote a completion of $\mathcal F$ w.r.t. $p\in P$ and ...
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1answer
34 views

Weak convergence of distribution family

I know the convergence in distribution and the weak convergence. but I have two questions: First one: does weak convergence implies pointwise convergence or it is the same? And second one: I have ...
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0answers
32 views

What is $f(x|y<\bar{y})$ equal to? [closed]

Where $x$ and $y$ follow a bivariate normal distribution, $x$ and $y$ are not independent, and $f()$ is a probability density function.
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22 views

probability theory books

since I'm going to deepen my knowledge of math stats I was wondering what book I should start from...I need one that covers in the very fine details the following topics trasformations among ...
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0answers
24 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
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0answers
12 views

To show that if T is a Weibull then aT also have a Weibull distribution [on hold]

(a)Show that if $T$ is a Weibull then $aT$ also have a Weibull distribution (b)$S(aT)$ is also Weibull Survival function
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0answers
26 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
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1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
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26 views

Maps that preserve Brownian motion law

I am looking for a list of maps that take Brownian motion to Brownian motion: Here are some: Any rigid transformation ...
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4answers
122 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
2
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1answer
53 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
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2answers
31 views

An $L^1$ convergence problem

Is the following true? If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$? ...
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1answer
420 views

Invariance of the correlation coefficient under linear transformations [on hold]

Show that for arbitrary random variables $X$ and $Y$, and constants $a,b,c,d$ with $a$ and $c$ nonzero, $$ \mathrm{Corr}(aX+b,cY+d) = \begin{cases} \mathrm{Corr}(X,Y)\quad&\text{if }a \text{ and ...
2
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1answer
25 views

Finding the probability of ever visiting a transient state for a zero-seeking device for a Markov Chain?

A zero-seeking device operates as follows: if it is in state $j$ at time $n$, then at time $n+1$, its position is $0$ with probability $\frac{1}{j}$ or $k$ with probability $\frac{2k}{j^2}$, where $k$ ...
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2answers
250 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
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0answers
18 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
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2answers
31 views

Resolvent operators and inverses proof

I am trying to prove for myself that $A(R_{\alpha}g)=\alpha R_{\alpha}g-g$ which is proving problematic. The definition of $A$, the generator, is $\displaystyle Af(x)= \lim_{t \rightarrow 0} ...
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1answer
46 views

Questions about expectation of stochastic integrals

I am considering the following SDEs: $$dX_1=-\theta(X_1-a_1)dt+\sqrt{X_1}(1-X_1)dW_1-X_1\sqrt{X_2}dW_2$$ $$dX_2=-\theta(X_2-a_2)dt-X_2\sqrt{X_1}dW_1+\sqrt{X_2}(1-X_2)dW_2$$ Here $W_1$ and $W_2$are ...
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0answers
24 views

Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
4
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2answers
84 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
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2answers
115 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
3
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1answer
111 views

Averaging inverse CDFs

Suppose I have two distributions $P$ and $Q$ on the line that admit well defined inverse cumulative distribution functions $F^{-1}_P$ and $F^{-1}_Q$. I define an "average" distribution $A$ as the ...
3
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2answers
33 views

Always null recurrence at the boundary between positive recurrence and transience?

I have the following theorem: Let $\rho$ be the traffic intensity. a) If $\rho<1$, then $X$ is positive recurrent. b) If $\rho>1$, then $X$ is transient. c) If $\rho=1$, ...
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0answers
20 views

Convergence of Beta Distribution to Bernoulli Distribution [on hold]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
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1answer
37 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
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0answers
56 views

Random Variables Proof Help [on hold]

I have two random variables $X$ and $Y$. Each of them having the conditions of Markov´s Inequality. So $X$ and $Y$ take on only nonnegative values, then for any value $a > 0$ $$P\{X \ge a\} \le ...
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1answer
156 views

Convergence in probability and almost surely

Let $X_n$ be a sequence of independent random variable which converges in probability to $X$. Prove $X$ is a constant. Can someone give me a hint how I should go about proving this? I tried proving ...
4
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1answer
105 views

Angle between $(X,Y)$ and $(E(X), E(Y)) $ where X and Y are independent random variables.

Suppose that $X$ and $Y$ are two independent random variables with known (different/same) probability distribution functions. Now consider the vector $(X,Y)$, I want to find the angle between $(X,Y)$ ...
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1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
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2answers
87 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
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2answers
974 views

How to split an integral exactly in two parts

This question is a by-product of a conversation with Theo Buehler in comments to this answer. Let's settle definitions. Definition Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. We say that ...
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0answers
16 views

Sequence of independent RVs [duplicate]

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant almost everywhere.
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0answers
69 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
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0answers
38 views

Can a biased physical random source be post-processed to control the bias?

Let $X_i$ with $i\in\mathbb N$ be a sequence of independent 6-ary random variables with distribution $\operatorname{Pr}(X_i=e)=p^e_i$ where $e\in\{1,2,3,4,5,6\}$ and $\sum_{e=1}^6p^e_i=1$. Let's ...
-1
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0answers
28 views

What is meant here with $\omega _k \in A _j=A_{n_k } $ [on hold]

I'm trying to understand what is meant by the following from Borovkovs probability theory. "Denote by $n_k $ the number of events $A_j $ such that $\omega _k \in A _j=A_{n_k } $, $n_k =0 $ if $\omega ...