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 ...

learn more… | top users | synonyms (1)

1
vote
1answer
22 views

$n$ dice, finding $\operatorname{var}(X)$

If we throw $n$ dice. And $X$ is the total number of eyes. Find $\operatorname{var}(X)$. My idea was to label $X=X_1+\cdots+X_n$ where $X_1$ is the outcome of die $1$ etc. And because $X_1,\ldots, ...
1
vote
0answers
21 views

maps of independent RVs when there is only pairwise independence

I know that for independent RVs $X_1, ..., X_n$ and measurable $g$, $h$, you have $g(X_1, ..., X_m)$ and $h(X_{m+1}, ..., X_n)$ are independent. But if there is only pairwise independence, that $X_i, ...
3
votes
2answers
56 views

Uniform convergence in distribution

Consider a sequence of stochastic processes, $X_n(x)$ and a limiting process $X(x)$. For a fixed $x$, if $\mathbb{P}(X_n(x) \leq y)$ converges to $\mathbb{P}(X(x) \leq y)$ for continuity points of ...
-1
votes
1answer
30 views

Number of ways in which four boys and four girls sit alternately in a row and one boy and one girl are not to sit in adjacent seats

Find the number of ways if four boys and four girls sit alternately in a row and one boy and one girl are not to sit in adjacent seats. I tried to get number of possible ways to sit alternately which ...
2
votes
2answers
37 views

pairwise independence implies independence for the multivariate normal distribution

Suppose that $(X_1, \ldots, X_N) \sim \mathcal{N(\bf{\mu}, \bf{\Sigma})}$ is multivariate normal, then I've seen that pairwise independence implies independence (wikipedia). How do you prove this ...
0
votes
0answers
35 views

Stopping times and expectation for the symmetric random walk

Let $X_n : \Omega \to \{ -1, 1 \}$ be a random variable with $P(X = -1) = P(X = 1) = 1/2$, like tossing a coin, and $M_n = \sum_{i=1}^n X_n$. Also let $\tau_m : \Omega \to \mathbb N \cup \{ \infty\}$ ...
0
votes
1answer
20 views

Randomization and probability with constraints

The probability of getting a specific suit out of a deck is 13/52. Once we have one suit (color) selected the probability is 12/51 for that suit and 13/51 for others. This much is obvious. Now ...
0
votes
0answers
18 views

Multiplying a non-central chi square distributed variable by a constant value

Let $X \sim \chi_k^2(\lambda)$, that is a non-central chi square distributed random variable with $k$ degrees of freedom and non-centrality parameter $\lambda$. What is the distribution of $Y=cX$, ...
0
votes
0answers
48 views

Almost sure convergence, product of variables

Consider the sequence of random variable given by distribution : $$\mathbb{P}(X_n=1)=1-\frac{1}{n},$$ $$\mathbb{P}(X_{n}=0)=\frac{1}{n}$$ and $Y_n=X_n \cdot Y$ for random variable Y. Does the $Y_{n}$ ...
0
votes
1answer
49 views

A dice and Law of Large Numbers

The dice A has 2 red faces and 4 green, and the dice B conversely: 4 red and 2 green. We toss a symetric coin; if come up heads, we choose the dice A, otherwise - the dice B. Next we execute a series ...
1
vote
0answers
20 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
0
votes
0answers
18 views

Sum of two gamma-distributed random variables

Let $X_1 \sim \Gamma(\alpha,\beta_1)$ and $X_2 \sim \Gamma(\alpha,\beta_2)$, what is the distribution of $Y = X_1+X_2$? Can it be expressed in terms of a certain probability distribution? Thanks
0
votes
0answers
23 views

Doob's optional sampling theorem

Say we have a right continues super-martingale $(X_t)_{t\geq 0}$ with filtration $F_t$ and a stopping time $\tau$ for which $P( \tau < \infty)=1$ why is it true to claim that $(X_{\tau\wedge ...
0
votes
0answers
37 views

How can we deduce uniqueness for SDEs by Girsanov's theorem?

Let $\mu\in L^{\infty}(\mathbb{R};\mathbb{R})$ be a bounded deterministic function. Then my understanding is that by using Girsanov's theorem, we can deduce uniqueness (in law) for the following ...
0
votes
1answer
48 views

Almost sure convergence of random variables

$(X_n)$ is a sequence of random variables having the following distribution: $$P(X_n=1)=1- \frac{1}{n},\; P(X_n=0)=\frac{1}{n}$$ (we don't assume that those variables are independent). $X$ is some ...
-1
votes
1answer
24 views

Probability question about sharing range of quantity

Suppose we have 10 Boxes, John shares into 7 of them, and Mike shares into 5 boxes. **The Question :**what is the expected number of boxes are shared between John and Mike? using equation :7*5 /100 ...
0
votes
1answer
33 views

Find the mean and variance of $V_n=\frac{1}{n}\sum_{i=1}^n(X_i-u)^2$

Suppose that $X_1,X_2,...,X_n$ is a random sample from a distribution with mean $\mu$ and variance $\sigma^2$. Suppose also that $v:=\mathbb{E}[(X_1-\mu)^4]<\infty$. Find the mean and variance of ...
0
votes
1answer
22 views

Conditional expectation, cartesian product

Let $\Omega=X\times Y$ be a sample space of pairs $(x,y)$. Let $\zeta$ be a random variable on $\Omega$. How would you interpret notation $\mathbb{E}(\zeta \mid x)$? Does this have any sense? I've ...
2
votes
1answer
20 views

Consider joint p.d.f. $f_{X,Y}(x,y)=C_1e^{-x-y}$, where $0<x<y<\infty$. Find $C_1$

Consider joint p.d.f. $f_{X,Y}(x,y)=C_1e^{-x-y}$, where $0<x<y<\infty$ with continuous random variables $X$ and $Y$. Find $C_1$. Not sure how to approach this. The main property that I am ...
0
votes
1answer
33 views

Sum of weighted chi square distributions

Let $X_1 \sim \chi_{k}^2$ and $X_2 \sim \chi_{k}^2$ (both i.i.d) and $a_1$ and $a_2$ positive real values. How can be expressed the PDF of $Y = a_1X_1 + a_2X_2$? Is it also a chi-square distribution? ...
0
votes
1answer
53 views

Convergance of some series of random variables

Random variables $(X_{n})$ are independent and have the distribution $P(X_{n}=1)=p, P(X_n=-1)=1-p$, $\frac{1}{2}<p<1$. Prove that $$X_1+X_2+\dots+X_n \to \infty $$ almost sure. Let ...
2
votes
0answers
31 views

Convergence of probabilties with indicator function

Let $\{p_j\}_{j = 1}^{\infty}$ be a set of nonnegative values such that they all sum to $1$, let $\mathcal{B}$ be a $\sigma$-algebra of subsets of the countable sample space $S = \{s_1, s_2, \dots\}$, ...
3
votes
2answers
54 views

Show that $\frac{X_1+\dots+X_n}{n}$ converges to $\infty$ a.s. for $X_n \sim U([0,n])$ independent

Random variables $(X_{n})$ are independent and $X_{n}$ has an uniform distribution on $[0,n]$ for n=1,2,... Prove that: $$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}\rightarrow \infty$$ almost sure. We can ...
1
vote
0answers
13 views

Why the marginal distribution function F_x(x) = F_xy(x,∞)?

I don't know if the next reasoning is all right. If the joint distribution function $F_{xy} = (\infty,\infty)$ = $ P(-\infty<X<\infty,-\infty<Y<\infty)$. That's the probability of the ...
2
votes
1answer
41 views

Data that follows a distribution with non-finite expectation

I find it difficult to get around the idea of some random variables following some distributions (such as the Cauchy Distribution) not to have finite means. How does one actually interpret data from ...
3
votes
1answer
26 views

Why is the probability different? We launch two dice

We launch two dice with faces $1 , 2, \dots, 6$, one black, one white. I want to calculate the probability that the sum of the two dice's faces is $\geq 11$. If I choose the main set as $\Omega = \{ ...
2
votes
2answers
59 views

Show that $\frac{X_1+\dots+X_n}{n}$ converges a.s. for $X_n \sim U([0,1-2^{-n}])$ independent

Let $(X_{n})$ be a sequence of independent random variables and let $X_{n}$ have a uniform distribution on $[0, 1-2^{-n}]$. Prove that the sequence: $$\frac{X_{1}+X_{2}+\dots+X_{n}}{n}$$ converges ...
0
votes
0answers
24 views

Buffon noodle problem gives theoretical issues

Let $\Gamma$ be a rectifiable curve in plane, having length $l$. Denote by $X_{\Gamma}$ the random variable that represents the number of crossings between $\Gamma$ and a grid of $d$-spaced parallel ...
1
vote
0answers
34 views

expectation value of independent random variables

In the statistics lecture that I'm attending, the professor once used the following: $X, Y$ random variables and i.i.d., then $$\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$$ I was trying to see an ...
0
votes
1answer
37 views

Is every continuous local martingale a uniform limit of step-processes?

The following question pertains to Wengenroth's textbook "Wahrscheinlichkeitstheorie", de Gruyter 2008 (in German). The covariance (aka compensator) of the continuous local martingales $X, Y \in ...
0
votes
1answer
28 views

Probability of losing everything in N games

Consider a gambler who starts with an initial amount of money of $£i$, obtains $£R$ with probability $p$ and loses $£J$ with probability $q=1-p$. What is the probability that it loses everything if he ...
1
vote
2answers
57 views

Infinite intersection of an interval and probability of selecting a random point

I am attempting to solve the following: Let $A_n= (\frac{1}{2}-\frac{1}{2n}, \frac{1}{2}+\frac{1}{2n})$. Show that $\bigcap\limits_{n=1}^\infty A_n=\{\frac{1}{2}\}$. Then apply the continuity ...
0
votes
2answers
33 views

A simple probability inequality

I am looking for a simple proof of the following inequality. Suppose that $X$ and $Y$ are two random variables. For any $x>0$ and $y>0$, $$ \Pr\{|X+Y|\ge x\} \ge\Pr\{|X|\ge x+y,|Y|\le y\}. ...
2
votes
1answer
44 views

ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence ...
3
votes
2answers
83 views

Probability of get a kidney donated

It is really a probability problem. I use the story of kidney donation because it is easier to describe. Consider the following scenario: Time is discrete. At each period, the measure of patients ...
2
votes
0answers
30 views

Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
2
votes
1answer
31 views

Buffon needle - variation

We have a paper ruled with equally space lines of distance $d$. Let $X_l$ be the random variable which represents the number of line intersections for a given needle of length $l$. Recall that the ...
1
vote
1answer
27 views

Ito's Isometry using Brownian Motion

Let $B_t$ be standard Brownian Motion. Could someone please help me to show that $$E[(\int_{0}^{t}B_sdB_s)^2] = \int_{0}^{t}E[B_s^2]ds$$ I am sure that it has something to do with Ito's formula but ...
1
vote
0answers
30 views
+50

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
1
vote
0answers
67 views

Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...
1
vote
1answer
30 views

How do I show that $P(|X-Y|>1/n)=0 \forall n \in \mathbb{N}$ then $X=Y$ a.s

How do I show that if $P(|X-Y|>1/n)=0 \forall n \in \mathbb{N}$ then $X=Y$ a.s Though this looks fairly obvious but how do i put it down rigorously . Can I say that $P(|X-Y|>0)=P\big(\bigcap_{n ...
2
votes
1answer
39 views

Prove that $X_n\cdot Y_n \to a\cdot X$ in distribution

Problem: Let $X, X_n, Y_n\, (n\in \Bbb{N})$ be random variables such that $X_n \to X$ in distribution and $Y_n \to a$ in probability for some $a\in \Bbb{R}$. Then $$X_n\cdot Y_n \to a\cdot X ...
2
votes
2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
1
vote
0answers
26 views

Induced probability measure for argmax

Let us consider a metric space $S$ consisting of random functions $f:[0,1]\rightarrow \mathbb{R^+}$ and a probability measure $P$, defined on this space. Now suppose we define a new function $$\hat ...
0
votes
3answers
61 views

Isn't $\mathbb{P}$ already a probability measure, so what is there to prove?

Follow-up to Probability measure over finite sample space. This is a theorem from Casella and Berger's Statistical Inference: Let $S = \{s_1, \dots, s_n\}$ (sample space) be finite and $p_1, ...
0
votes
0answers
29 views

Extension of Chebyshev's inequality

Consider about the sum of iid random variables $$ X_t = \sum_{i=1}^{t}x_i $$ $x_t$ are of zero mean and finite variance $\sigma^2$. Let $M,m>0$. By Chebyshev's inequality, it is easy to show that ...
0
votes
0answers
25 views

Probability of hitting zero

Suppose time is discrete. $X_{t+1} = X_t + x_t$. $x_t$ is of continuous value, iid with mean zero and finite variance. Let initial condition $X_0>0$, how can I prove that the probability of $X_t$ ...
1
vote
1answer
39 views

Why isn't $\mathcal D$ a sigma-algebra?

I came across the statement that if $(\Omega, \mathcal F, \mathbb P)$ is a probability space and $E \in \mathcal F$ then $$\mathcal D := \{ A \in \mathcal F \mid A \text{ and } E \text{ are ...
1
vote
1answer
39 views

Construction on Ito Integral with Brownian Motion

I have just started learning stochastic calculus and my professor posed the following as exercises to help understand how we construct the Ito Integral. Let $B$ be a standard Brownian motion. Fix ...
3
votes
2answers
36 views

Given the p.d.f. of X, find the mean and variance of $X$.

Suppose that $X$ has probability density function, $$f(x)=\begin{cases} \dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1} & \text{for }0\leq x\leq1 \\[8pt] 0 ...