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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Expected number of lines in use in call centre (markov process: queuing theory)

Suppose we have a call centre with infinitely many lines to be able to call to. Calls come in a rate of $\lambda$ and customers are served with rate $\mu$. It is easy to see that the $Q$-matris looks ...
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22 views

$\chi_q=\lambda t E[Y_{1}^q],$ $\psi(u,t)=\exp[-\lambda t (1-\psi Y_{1}(u))]$ is a compound Poisson process has the second moment properties . [on hold]

Let $C(t)=\sum_{n \ge1}Y_{n}1_{\{t\ge T_{n}\}}=\sum_{n=1}^{N(t)}Y_{n}$ is compound Poisson process. Show that a compound Poisson process has the second moment properties in $E[C(t)]=\lambda tE[Y_1],$ ...
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0answers
26 views

If $E[X(t)X(s)]=t \land s $. Show that this process has independent increments [on hold]

Let $X(t), t\ge0$ be a real-valued Gaussian process with zero mean and the covariance function $\mathbb{E}\left[X(t)X(s)\right] = t \land s $. Show that this process has independent increments.
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26 views

probability of hitting state $i$ in random walk

We have a random walk on the integers with probability of going to the right is $\lambda$ and to the left is $\mu$. Suppose we start at 0. I want to find the probability of ever hitting a fixed state ...
0
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1answer
38 views

Distribution of transformed random variables

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution ...
3
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2answers
32 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
2
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1answer
48 views

Compute almost sure limit of martingale?

Let $Y_1, Y_2, \dots$ be nonnegative i.i.d random variables with mean 1. Let $$X_n = \prod_{m \le n}Y_m$$ If $P(Y = 1) < 1$, prove that $\lim_{n->\infty}X_n = 0$ almost surely. I feel like ...
2
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2answers
40 views

Limit law of real-valued independent random variables

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp. i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$. Then, are $X$ ...
0
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1answer
65 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
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31 views

Weak Law of Large Numbers for asymptotically uncorrelated random variables [on hold]

$X_n$ is a sequence of random variables with $Var(X_n)\le{c} \space \forall \space n$ where $c \in (0,\infty)$ and $$Corr(X_i,X_j)\rightarrow 0 \space \space \text{if} \space \space ...
2
votes
1answer
26 views

distribution of the difference of discrete uniform RVs

Let $P_1, P_2$ be independent discrete uniform random variables on $\{0,1,...,k\}$. Suppose we want to compute $$\mathbb{P}(P_1 > P_2).$$ Is the best approach to see $\mathbb{P}(P_1 > P_2) = ...
2
votes
2answers
38 views

St.Petersburg Paradox and Bernoulli's quote

I was reading about St.Petersburg paradox, and understood the proof that $\frac{S_n}{n\log n} \overset{P}{\rightarrow}1$. The textbook then quotes Bernoulli: "There ought not to exist any even ...
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3answers
53 views

Calculate expected value. [on hold]

Can someone give a hint for v). I don't know how to evaluate this integral from 0 to infinity. Thank you!
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0answers
29 views

Using gamma distribution to find the average duration of breaks after 10 calls with exponential distribution

Worker works 8 hours a day. Time between $ 2$ calls has $\exp(4)$ distrubution (expecting $4$ calls per hour). Duration of calls is $0$ (he just registers them). After $10$ calls he goes to $15$ ...
2
votes
2answers
42 views

Limiting probability that the sum of the values of a die is a multiple of 13

A fair die is thrown repeatedly. Let $X_n$ denote the sum of the $n$ first throws. I have to find $\lim_{n\rightarrow \infty}P(X_n \text{ is multiple of 13})$. Now follows what I tried, which I don't ...
0
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1answer
28 views

If $X^T(t)=X(t\land T)$ is said to be the process $X$ stopped at $T$. I want prove following statment

Let $X$ be a stochastic process defined on a probability space $(\omega ,\mathcal F,P)$ endowed with a filtration $(\mathcal F)_{t \ge0}$ and let $T$ , $T^\prime$ be $\mathcal F_{t}-$stopping times. ...
6
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2answers
121 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 ...
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1answer
14 views

Show that the expectation of a submartingale and supermartingale is an increasing and decreasing function of time, respectively.

Show that the expectation of a submartingale and supermartingale is an increasing and decreasing function of time, respectively. thanks for help
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2answers
24 views

sum of two random variables with geometric distribution

hello. i've got 3 random variables, $X$, $Y$ with $GEO$ ~ $(p)$ for both , and $X+Y = Z$. i need to calculate $P(X | Z=k)$. so i started with: $P(X|Z=k)=P(X|X+Y=k)=P(x=j|X+Y=k)= ...
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1answer
35 views

A proof about martingales and variance

We consider a martingale $(S_n)$ with $\mathbb E(S_n^2)<K<\infty$. Suppose that $\mathrm{ Var}(S_n)\rightarrow0$. Prove that $S=\lim_{n\rightarrow \infty}S_n$ exists and is constant a.s. I ...
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2answers
43 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
2
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0answers
32 views

What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion?

Let $S= \{x \in [0,1 ]: \lim_{n \to \infty } \frac {1 } {n } \sum _{i=1 } ^n d _i (x) = 2/3 \} $, where $d _i(x) $ gives the ith digit of the infinite base-2 expansion of $x $. What is the Lebesgue ...
2
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2answers
58 views

Simplified Strong Law of Large Number by Using Truncating Function

Consider $X_1,X_2,...$ be i.i.d. random variables with $E|X_i| <\infty$ and let $EX_i := \mu$ and $S_n := \sum_{i=1}^n X_i$. Now, consider the corresponding truncated random variables $Y_k := ...
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2answers
31 views

Probability: Random Sample Problem

I need some help on the following problem: Let $X_1$ and $X_2$ be random sample from the pdf \begin{equation} f(x) = \begin{cases} 4x^3,&0<x<1\\ 0, & \text{otherwise} \end{cases} ...
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0answers
35 views

Undergraduate Probability

I have a general question about how one goes towards teaching undergraduate probability theory in a careful way. I remember when I took probability theory as an undergraduate I completely despised it. ...
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0answers
33 views

probability and bernoulii random variable? [closed]

A database file has 6,000,000 (six million) records, which occupy disk storage at a density of 12 records per block. A weekly update modifies 6.5 percent of the file and we assume that the changes ...
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Using Bayes' Theorem,compute the probabilities [closed]

A manufacturing process produces computer chips of which $6$ percent are defective. This percent is actually found using a thorough (and expensive test) on a small random sample of chips. The plant ...
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0answers
39 views

Random Codebook Generation

I do generate a random codebook $\mathcal{C}$ by generating $2^{NC}$ codewords $X^N=[X_1\;X_2\;\cdots\;X_N]$ randomly and independently, each according to some distribution $p_{X^n}(x^n)=\Pi_{i=1}^n ...
0
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1answer
23 views

4 vehicles Probability that any three of them are operational at one time [closed]

I have annual vehicle operational availability for 4 vehicles (A-75%, B-85%, C-90%, D-90%). I am trying to figure out what the probability is that at any time $3$ of the $4$ are operational. Any ...
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0answers
26 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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1answer
31 views

Finding the Cumulative Distribution Function

the solution from my teacher is as follows; My question is, for $0 \le x \le 1$, why do we have instead of just having ? Does it have something to do with the discontinuity of the density ...
3
votes
1answer
37 views

A property about tail equivalent random variables

Let $(X_n)$ and $(Y_n)$ be tail equivalent random variables i.e. $\sum_{i=1}^{\infty}\mathbb P(X_i\neq Y_i)<\infty$ Show that $\sum_{n=1}^{\infty}X_n$ and $\sum_{n=1}^{\infty}Y_n$ converge or ...
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20 views

Can SLLN be applied to $Y=\frac {1 } {k} \sum _{n=1 } ^ {k} Z _n 2^{-n } $, where $Z _n $ are iid?

Can the strong law of lagre numbers be applied to conclude convergence of sum $\frac {1 } {k} \sum _{n=1 } ^ {k} Z _n 2^{-n } $, where $Z _n $ are iid?? I can see that every partial sum can be ...
0
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0answers
52 views

First-hitting probability for the 2D critical site percolation on triangular lattice

Consider critical site percolation on the triangular lattice with mesh-size $\delta$. Let $D$ be the domain $H\backslash[0;i]$ where $H$ is the complex upper half plane. Compute the probability $$ ...
5
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2answers
50 views

probablity that $\max(X,Y)> a \min(X,Y)$

Two independent random variable $X$ and $Y$ having probability density functions uniform in the interval [0,1]. when $a \geqslant 1$, the probability that $\max(X,Y)> a \min(X,Y)$ is? (in terms of ...
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1answer
39 views

Brownian motion is almost surely continuous

Why is Brownian motion required to be almost surely continuous instead of merely continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
1
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1answer
28 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
2
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1answer
70 views
+50

If a joint cdf is increasing in each argument, then the pdf is strictly positive a.s.?

Let $F:\mathbb{R}^d \to [0,1]$ be an absolutely continuous joint cdf and let it be strictly increasing in each argument. Does it imply that its pdf $f$ is strictly positive a.s. (with respect to the ...
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0answers
14 views

What are the assumptions for applying Wald's equation with a stopping time

I am trying to understand the assumptions under which I am allowed to apply Wald's equation for a sum of a random number $N$ of random variables $X_n$, $1\leq n\leq N$. There seem to be several ...
2
votes
2answers
47 views

Sample path of Brownian Motion within epsilon distance of continuous function

Given a continuous function $f:[0,1]\rightarrow\mathbb{R}$, $f(0)=0$, how can one show that $P(\underset{0\leq t\leq1}{\sup}\left|B_{t}-f(t)\right|<\varepsilon)>0$, where $P$ is the probability ...
1
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1answer
34 views

A queueing model issue.

I am very beginner in Queueing Theory and I am learning in my own. I am struggling in the following situation. Suppose in a service center if a job arrives it will immediately being processed if a ...
0
votes
1answer
28 views

Exponential(1) distribution of Normally distributed X and Y

Let $X_1,X_2,X_3,X_4,X_5$ be a random sample from the uniform pdf: $f(x)= 1$, $0<x<1$ zero otherwise. Show that $\ln X_i$ has Exponential($1$) distribution for $i=1,2,3,4,5$. Solution: Let ...
0
votes
1answer
16 views

Mean and Variance of Nornally distributed distribution

Given X and Y be jointly normally distributed with $\mu_x=20, \mu_Y=40,\sigma_x=3, \sigma_Y=2$ and $\rho=0.6$. Find the mean and the variance of U=X+Y. soln: $U~N(\mu=60,\sigma^2=13). Am I right?$
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Sequence $\frac{x_{n}}{n} \sim Bin(1, \frac{1}{n^{2}})$. Show $x_{n}$ converges to $0. [closed]

Sequence $\frac{x_{n}}{n} \sim Bin(1, \frac{1}{n^{2}})$. Show $x_{n}$ converges to $0. Please help me with it, thanks very much
2
votes
1answer
33 views

Find $\operatorname{Corr}(XY,Y)$ and $\operatorname{Corr}(X^{2},Y^{2})$

$\def\Cov{\operatorname{Cov}}\def\Corr{\operatorname{Corr}}\def\Var{\operatorname{Var}}$ Suppose that $X$ and $Y$ have a joint normal distribution with $E(X)=E(Y)=0, \Var(X)=\Var(Y)=1, ...
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3answers
35 views

Probability question - arranging 20 pupils in a row - 8 boys and 12 girls

We have 20 pupils in class, 12 girls and 8 boys. We arrange the pupils in a row, and now need to calculate the following probability: a. The probability that Jana, one of the girls, will not stand ...
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3answers
44 views

Probability question - arranging 20 pupils in a row

Will someone help me figuring the following out ? We have 20 pupils in class, 12 girls and 8 boys. We arrange the pupils in a row, and now need to calculate the following probability: The boys will ...
1
vote
1answer
23 views

Union Bound for a Random Number of Events

Is it possible to generalize the union bound to a random number of events, similar to Wald's equation about a random number of terms in a sum? In particular, how can I bound something like $$ ...
0
votes
2answers
41 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
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0answers
31 views

Amalgamated product of two measures.

Please do not get annoyed by the symbols below. The problem has a really simple statement. $2^X$ denotes the set of functions from $X$ to $\{0, 1\}$ equipped with usual product topology. By $Fn(X, ...