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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0answers
12 views

Markov Chain States

Given a MC with five states $\{1,2,3,4,5\}$ and transition matrix \begin{bmatrix} 0.5& 0.5 & 0 & 0 & 0 \\ 0.75 & 0.25 & 0 & 0 & 0 \\ 0 & 0.25 & ...
1
vote
0answers
8 views

dependent “time change” of a.s. convergent random variables

Let $(X_n)$ be a sequence of random variables, s.t. $\frac{X_n}{n^p}\to X$ a.s. for some $p>0$. Now let $(Y_t)$ be a discrete stochastic process, s.t. $\frac{(Y_t)^p}{t}\to Y>0$ a.s. We only ...
1
vote
1answer
7 views

Augmentation of a Filtration

In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a ...
1
vote
1answer
15 views

Poisson process, martingale

I am asked to show that if N is a poisson process of intensity 1, then: (I) $X_t=N_t-t$ is a martingale (ii) $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
11
votes
2answers
166 views

Using probability methods prove $\frac{\sin(t)}{t} = \prod_{i=1}^{\infty} \cos \left( \frac{t}{2^i}\right )$

Using probability methods (characteristic function?) prove $$\frac{\sin(t)}{t} = \prod_{i=1}^{\infty} \cos \left( \frac{t}{2^i} \right)$$ I know what is characteristic function but I have no idea ...
2
votes
0answers
28 views

Random Walk Question - what is the probability of eventually reaching the origin? [duplicate]

Consider the random walk $S_n$ given by $S_{n+1} = S_{n} + 2$ with probability $p$ ; $S_{n+1} = S_{n} - 1$ with probability $1-p$. Assume that $S_0 = n > 0$ with certainty. What is the ...
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0answers
22 views

Is this Markov Chain irreducible? Aperiodic? What is its equilibrium mass function? [on hold]

Consider a Markov chain with outcomes $\{0,\dotsc, n\}$ and transition probabilities \begin{align*} P_{i,i+1} &= p \\ P_{i,i-1} &= q \end{align*} for $1\leq i \leq n-1$ and $p+q=1$. Assume ...
1
vote
1answer
19 views

Setting up the expected value for $x_t=\sin(2\pi U t)$.

We have the series $x_t=\sin(2\pi U t)$ where $t=1,2,3,\ldots$ and $U$ is uniform on the interval $(0,1)$. I have to find the expected value of $x_t$. I always thought that if $X$ is a continuous ...
-1
votes
0answers
14 views

Show the equilibrium vector of a transition matrix for a Markov Chain has no zero entries [on hold]

Let P be a transition matrix for a regular Markov chain and let w be its equilibrium vector. Show that w has no zero entries.
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0answers
34 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
3
votes
0answers
38 views
+50

solution uniqueness of an algebraic system

$A(v),B(v),C(v)$ are positive, convexly decreasing functions on $\mathbb{R_+}$; $x$ is a random variable that obeys distribution F; Function $v(a,b,x)$ is implicitly defined as ...
0
votes
1answer
1k views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
2
votes
0answers
12 views

Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...
0
votes
0answers
7 views

Calculating variance & expected value of a statistic with exponents

I am trying to calculate of a statistic: $Var(\frac{1}{1 + 1/n \sum_i x_i})$. Thus far, I have $=E[(1 + 1/n \sum_i x_i)^{-2}] - E[(1 + 1/n \sum_i x_i)^{-1}]^2$. How do you deal with exponents inside ...
0
votes
0answers
15 views

Mathematical statistics: Pollen dispersal directionality

What Information am I looking for? Think about a tree that is sending pollen all over the place. Because of wind, most pollen grain will go toward one direction. Imagine, we split the 2D area around ...
0
votes
1answer
25 views

Definition of an absolutely continuous random variable

Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like: $$\mathbf{P} (A) = \int_A f d \mu$$ for some Borel set $A$. But what is $\mu$? Is ...
1
vote
1answer
9 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
-1
votes
1answer
27 views

Geometric Brownian Motion [on hold]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...
3
votes
1answer
30 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
2
votes
1answer
53 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
0
votes
1answer
51 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
2
votes
0answers
12 views

UMVUE for altered Normal distribution

Let $X_1 , ...,X_n$ be a sample from a normal population $N(\mu , \sigma^2)$. It's easy to find the UMVUE for $\mu$ and $\sigma^2$: (1) After finding the joint density of X=$(X_1 ,...,X_n)$, we find ...
4
votes
0answers
22 views

Analysis of sorting Algorithm with probably wrong comparator?

It is an interesting question from an Interview, I failed it. An array has n different elements [A1 .. A2 .... An](random order). We have a comparator C, but it has a probability p to return correct ...
0
votes
0answers
21 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
2
votes
1answer
16 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
1
vote
0answers
26 views

Markov factorization of the density of an AR(1) process

Suppose we have a causal, stationary AR$(1)$ process with i.i.d. innovations $Z_t$. Then we know that it is a Markov as future value $X_{t+1} = \phi X_t + Z_{t+1}$ given the past $X_1,\ldots X_t$ ...
2
votes
1answer
318 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
4
votes
2answers
49 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
0
votes
0answers
18 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
3
votes
1answer
26 views

second Borel–Cantelli lemma (or converse result) application

I'm having issues with understanding how Borel-Cantelli lemma applies to the following exercise: If a coin is tossed infinite times, prove that the probability of getting 2 consecutive heads (or ...
8
votes
2answers
104 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
0
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0answers
26 views

Does the law of large numbers pin down the distribution of an infinite sample?

Imagine you draw (independently) an infinite amount of draws from a random variable with infinite support, and the strong law of large numbers applies. We know the average for sure will be equal to ...
2
votes
1answer
24 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
0
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0answers
8 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...
0
votes
1answer
17 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
1
vote
1answer
81 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
0
votes
0answers
15 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
7
votes
1answer
90 views

Show that $f(W_t)-\frac{1}{2} \int_0^t f''(W_s) \, ds$ is a martingale without using Itô's formula

I'm learning the basics about Brownian motion (I know nothing about stochastic calculus), and I've shown that if $W(t)$ is a standard Brownian motion, then $W(t)^2-t$ is a martingale. Now I'm trying ...
1
vote
0answers
19 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
1
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0answers
31 views

Expected value of conditional expectation, discrete variable

We are given a random variable $X$ on $\Omega_1$ and a discrete variable $Y: \Omega_2 \to \mathbb{N}$. We consider $\mathbb{E}(X|Y)$ as a random variable defined as follows: $$\mathbb{E}(X|Y)(\omega)= ...
0
votes
1answer
78 views
+50

Result of a $2D$ random walk with position dependent probabilities of step

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:- A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
2
votes
0answers
13 views

when is the maximum likelihood estimator measurable

For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood ...
4
votes
1answer
132 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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votes
2answers
44 views

Conditional Probability Question. [on hold]

A letter is known to have come from either 'TATANAGAR' or 'CALCUTTA'. On the envelop just two letters 'TA' are visible. What is the probability that the letter has come from (i) TATANAGAR (ii) ...
1
vote
1answer
35 views

Show $\mathbb{E}(X \mid Y,Z) = \mathbb{E}(X \mid Y)$ if $Z$ is independent of $X$ and $Y$

Let $X,Y,Z$ be random variables, $X$ integrable, $Z$ independent of $X$ and $Y$. Then we have $E[X\mid Y,Z]=E[X\mid Y]$. Why is only assuming $Z$ independent of $Y$ not enough. I was able to ...
0
votes
0answers
8 views

Interpretation of sufficient statistic in the continuous case

A statistic $S = S (X)$ is called sufficient for $\theta$ if there is a $P_{X \mid S} (\cdot \mid s)$ that doesn't depend on $\theta$. So if $S(X)$ is a discrete random variable and we know $S (X) = ...
2
votes
1answer
34 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
3
votes
0answers
26 views

Conditional expectation, sigma algebra

Let $X$ be a random variable on $\Omega$ and $Y$ a discrete variable having values $y_1, y_2,...$. We define another random variable via conditional expectation $\mathbb{E}(X|Y)(\omega) = ...
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votes
0answers
15 views

Random variables set representation in the sample space [on hold]

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
0
votes
0answers
35 views

Proving a sequence forms a martingale

Let $\Omega = \mathbb N = \{1,2,3,\cdots\}$ and $\mathscr F_n$ be the $\sigma$-field generated by the sets $\{1\},\{2\},\cdots,\{[n+1,\infty)\}$ Define a probability on $\mathbb N$ by setting ...