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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Proof of “changes of sign” in one-dimensional random walk model [Feller's section 3.5, page 84]

Consider the one-dimensional random walk of a particle. We shall denote the individual steps by $X_1, X_2, \cdots$ with $X_i = \pm 1$ and the positions by $S_1, S_2, \cdots$ with $S_i = X_1 + X_2 + ...
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34 views

If X and Y are equal almost surely, then they have the same distribution, but the reverse direction is not correct

Show that if two random variables X and Y are equal almost surely, then they have the same distribution. Show that the reverse direction is not correct. If $2$ r.v are equal a.s. can we write ...
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How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
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41 views

Let $Y,Z$ be idd. RV's, $\ Z \sim e(1), \ Y \sim R(0,1)$. Find $f_{X,Y}$ for $(X,Y)$.

Let $Y,Z$ be idd. RV's, $\ Z \sim e(1), \ Y \sim R(0,1), \ X = \frac Z Y$. ($R(0,1)$ denote the continuous uniform distribution) Compute $P(X>1)$: I have $P(X>1) = 1-P(X \le 1) = 1 - P(Z\le ...
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36 views

Let $Y,Z$ be independent stochastic variables, $Y \sim Z \sim N(0,1)$ and $X := Y+Z$. Find $f_{X,Y}$ for $(X,Y)$ using transformation theorem.

Let $Y,Z$ be independent stochastic variables, $Y \sim Z \sim N(0,1)$ and $X := Y+Z$. Find $f_{Y,Z}$ for $(Y,Z)$ and $f_{X,Y}$ for $(X,Y)$ using transformation theorem. Since $Y,Z$ are independent ...
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92 views

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval.

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval. In particular, show that the minimum length interval satisfies $f_{(n+3)}(a) = f_{(n+3)}(b)$, where ...
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44 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
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9 views

invariant measures for the independently coupled Feller process

If I take countably many Markov processes and couple them independently, how does the collection of invariant measures for the coupled process relate to those of the marginals? I conjecture that it ...
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10 views

Existence of density

Suppose $ \{ D_i : i \geq 1 \} $ be a sequence of i.i.d. random variables taking values $ \{ 0, 1, 2, \dotsc, K-1 \} $ where $ K \geq 3 $ is a positive integer with probabilities $ \mathbb{P} ( D_1 = ...
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43 views

show that MSE$(\hat{\theta}) = E[(\hat{\theta} − θ)^2] = V(\hat{\theta}) + (B(\hat{\theta}))^2$.

Using the identity $(\hat{\theta} − θ) = [\hat{\theta} − E(\hat{\theta})] + [E(\hat{\theta}) − θ] = [\hat{\theta} − E(\hat{\theta})] + B(\hat{\theta})$, I need to show that MSE$(\hat{\theta}) = ...
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15 views

Marginal Density Question

I am faced with the following question, which I think is quite simple, but I can't put together for some reason. Given that $f(x,y)=(6/5)(x+y^2)$ for $0<x,y<1$, ($f(x,y)=0$ everywhere else), I ...
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65 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose density function $ f $ is given by $$ f(y) = \begin{cases} ...
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24 views

How to represent?

You are a well-known hedge fund manager in Wall Street circles. One of your wealthy clients has $\$1$ million dollars to invest in XYZ stocks. Currently, XYZ stocks are trading at $\$2$ per share. You ...
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33 views

Shannon Entropy Minimization

The Shannon Entropy for an observation is given by $ -x \log_2(x)$. Why is the maximum entropy achieved at $x = \frac{1}{e}$, and not at $x = 0$? Could someone provide a logical explanation that ...
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19 views

How we compute expectation of a singular random variable?

In probability (or measure) courses, we often see the Cantor distribution that is singular with respect to the Lebesgue measure. Its CDF is increasing but whenever its differentiable, the ...
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37 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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throwing a dice repeatedly so that each side appear once. [duplicate]

Pratt is given a fair die. He repeatedly throw the die until he get at least each number (1 to 6). Define the random variable $X$ to be the total number of trials that pratt throws the die. or ...
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28 views

Equivalence, identity of random variables

Suppose I have $X \sim \text{Uniform}(0,1)$ and $Y \sim \text{Uniform}(0,1)$ As we all know $X+Y$ is a triangular distribution. What of $X+X$? Surely this is uniformly distributed on the interval ...
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56 views

Convergence of the series of identically distributed dependent random variables

Let $a_1$, $a_2$, $\ldots$ be identically distributed, positive, not necessarily independent random variables. Consider the series $$\sum^{\infty}_{n=1} a_n$$ Is it true that the series diverges ...
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22 views

Girsanov's theorem for OU process

Say that I've got the process $dr_t=a(b-r_t)dt+\sigma dB_t$ and that I want to calculate $E[\exp(-\int_0^T r_s ds)]$ by using Girsanov's theorem. How do I do this? I cannot seem to find an explicit ...
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34 views

Integrating a function of a random variable; $\int g(X) dP$

Assume a random variable $X$ on probability space $\Omega$, taking values in $\mathbb{R}$ with some known distribution $F(dX)$. Assume also a function of the random variable, $g(X)$. Does then the ...
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52 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
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38 views

Sum of probabilities or mean of probability

My question is about being confused about two way of approaching a problem, which in this case lead me to the same solution. One method is very verbose, the other one is fast and clean. Let's ...
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21 views

Random variables with random indexes

Suppose I have a pair of stochastic processes - $(X_n,I_n)$. Say $I_n$ is defined on $\mathbb{N}$. Both $I_n$ and $X_n$ may be Markov processes or iid random variables. I want to investigate the ...
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2answers
63 views

How many times does a single fair die have to be rolled for a number to repeat

PHP developer here so no real math background :-( gave up after googling and hence the question As the question states how many times does a single die have to be rolled for any of the previously ...
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1answer
62 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
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14 views

A question on limit of weak-* convergence of probability measures

Let $(X,\mu)$ be a measure space. Assume $X$ is compact. It is well-known that the space $\mathcal{P}(X)$ of probability measures on $X$ is compact in weak-* topology. Let's consider a sequence of ...
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The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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36 views

How to find PMF

A coin with probability p of heads is tossed until the first head occurs. It is then tossed some more until the first (subsequent) tail occurs. Let X be the total number of tosses required. Find the ...
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39 views

Kolmogorov equations: simple intuitive explanation

Can somebody give an intuitive explanation of the Kolomogorov forward and backward equations. Specifically in which situation you would use the forward or the backward equation.
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38 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
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An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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25 views

Let $(X,Y)$ be a random vector. Show $P(x,y) > 0$ implies $P(y) >0$ and $\sum_y P(x,y) = P(x)$ for $X = x, Y = y$ using the axioms of probability?

Let $(X,Y)$ be a random vector. How does one show $P(x,y) > 0$ implies $P(y) >0$ and $\sum_y P(x,y) = P(x)$ for $X = x, Y = y$ using the axioms of probability ? (In the continuous case the ...
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49 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
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49 views

Compute the density of $Y=|X|$

When $X$ has the normal distribution $\mathcal N(\mu,\sigma^2)$ , compute the density of $Y=|X|$ I know ...
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26 views

Explanation on one-dimensional random walk in Feller's book

Consider the random walk on the integer number line, $\mathbb{Z}$, which starts at 0 and at each step moves $+1$ or $−1$ with equal probability. The probability for the event that "the first return to ...
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34 views

Implications of convergence in probability?

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
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29 views

Characteristic function of the Binomial distribution converges to that of the Poisson

Find conditions on $\lambda, n, p$, so that the characteristic function of the Binomial converges to that of the Poisson Binomial distribution is given as $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ ...
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145 views

Convergence in probability to a non-measurable limit

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Denote the Borel field on $\mathbb{R}$ by $\mathcal{B}$. Let $\mu: \Omega \rightarrow [0,\infty)$ be a not-necessarily-measurable function and, ...
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factoring series of multiplication

Given that $y = \frac{1}{N} \sum_{i=1}^{N}\log_2(1+|x_i|^2 y z). $ I simplify the equation into $2^y= \frac{1}{N}\prod\limits_{i=1}^{N}\left(1+x_i^2(y)(z)\right)$ ...
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Draws from the uniform distribution are taken until the sum exceeds 1. What is the expected value of the final draw?

I was thinking about this question after a related problem: what's the expected number of draws for the sum to exceed 1? For that problem, the answer is known and is a surprising result ...
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Question about probability interpretation formula

Let $0<p<1$ y $0\leq k\leq n$. Prove that $$\sum_{n=k}^\infty\binom{n-1}{k-1}p^k(1-p)^{n-k}=1$$ I know how to prove this using mathematical analysis. But my probability teacher say that, there ...
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26 views

if the CDF is non-invertible or does not have a closed form solution(e.g. Normal CDF), how can we generate random data from such a distribution?

Given the CDF of a distribution to generate random data from that distribution by using the inverse transformation of the CDF. Then if the CDF is non-invertible or does not have a closed form ...
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27 views

Scale invariant measures must have power law densities

If $\mu$ is a scale-invariant measure(say on $\mathbb{R}^{+}$), i.e. for any set $A$, $\mu\left(\frac{A}{c}\right)=g\left(c\right)\mu\left(A\right)$ where $c>0$, then is it necessary that $g$ must ...
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18 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
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32 views

$f \le 1 \Rightarrow f =1 $ a.s.

I know the title doesn't say much, but I hope you'll help me nonetheless. Here's my problem. Let $P, Q$ be two probabilistic measures, $P$ is atomless and the measures have the same independent ...
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13 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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23 views

Relations between stochastic Big O-Notation, limsup and lim?

I have to solve a list of exercises on probability theory but I'm having several problems in understanding the following questions; could you help me? Thank you! Let $||.||$ indicate the Euclidean ...
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40 views

Law of large numbers variant?

I have the following: Let $(X_n)$ be a sequence of i.i.d. random variables. (a) Assume $\frac{1}{n} S_n=\frac{1}{n} \sum_{i=1}^n X_i$ converges a.s. to a real-valued random variable $Y$. Show that ...
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32 views

a martingale equality

Let $X_{t}$ a positive continuous martingale satisfying: $\lim_{t\longrightarrow \infty}X_{t}=0 $ ps and $X_{0}=a \in {R_{+}}$ Show that $\textbf{P}(\sup_{t\geq0}X_{t} \geq b)=\frac{a}{b}$ , a < ...