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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15 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
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0answers
7 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
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0answers
9 views

Exercise on stationary measures.

This is a question from Durrett, exercise 6.5.4. Recall that $$ \mu_x(y) = E_x\left( \sum_{n=0}^{T_x-1} 1(X_n = y)\right) = \sum_{n=0}^\infty P_x(X_n = y, T_x > n)$$ is a stationary measure and ...
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0answers
7 views

Asking for helps about deriving arcsine distribution

I solved the above exercise. And the exercise below is based on the exercise above. Here, I managed to show the first equality of (i). But I can't find a way how to prove the second equality of ...
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1answer
20 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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1answer
25 views

Finding Variance

I am a little confused on how to go about finding different parts of the Variance of a random variable. Here is the question. A total of $n$ balls, numbered $1,.. n$, are put into $n$ urns, also ...
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0answers
27 views

understanding darts probability

Note: this problem for who understands the game of darts Hello iam trying to compute the probability of a dart to hit a ring if you know that the opportunity to miss the ring is 10% what will the ...
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1answer
24 views

Divergent series of independent RV

I'm trying to prove that if $\{X_n\}_{n=1}^{\infty}$ is a sequence of independent random variables with the same distribution and $P(X_1 \neq 0)>0$, then the series $\sum_{n=1}^{\infty} X_n$ is ...
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0answers
8 views

Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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1answer
47 views

How to find $E[X^2\mid X+Y]$?

Suppose $X$ and $Y$ are independent Poisson random variables with rates $\lambda_1, \lambda_2$ respectively, then how would we go about calculating: $ E[X^2\mid X+Y] \text{ ?} $$
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1answer
15 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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1answer
18 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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0answers
23 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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1answer
20 views

Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
0
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1answer
20 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
1
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1answer
18 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
1
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1answer
42 views

How long would it take to a lottery number repeat?

In Professor Stewart’s Cabinet of Mathematical Curiosities the following is asked: You have $1000$ songs on your MP3 player. If it plays songs ‘at random’, how long would you expect to wait ...
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1answer
41 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
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1answer
28 views

New characteristic function from old

The question I want to do says: Let $f(u,t) : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function, such that for each $u$, $f(u, \cdot)$ is a characteristic function, and such that for each $t$, ...
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0answers
28 views

Probability_distribution [on hold]

Three points are chosen at random on the circumference of a circle. Find the probability that they all lie on the same semicircle, using random numbers generated from a uniform distribution.
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1answer
29 views

4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
2
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1answer
52 views

incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
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1answer
24 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
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1answer
23 views

Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
2
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0answers
20 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
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1answer
10 views

Bounds-negative binomial distribution

Suppose $Y=\sum_{i=1}^{n} X_{i}$ where each $X_{i}$ is an independently and identically distributed geometric random variable with success parameter $p$, so that $Y$ has a negative binomial ...
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1answer
29 views

Representing the probability as a recurrence equation

Introduction Suppose that you initially have an $n$-sided die with equal probability and you throw it then you will get a certain number $1< k \leq n$ then you throw a $k$-sided die. Continue ...
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0answers
17 views

conditional expectation of exponential random variable conditoned on sum of exponential random variable

Let X,Y be i.i.d. exponentially distributed with parameter $\lambda$. Show that for $Z:=X+Y$ and a measurable, non-negative function $h$ we have: ...
1
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1answer
14 views

Local martingale being true martingale

I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale. In ...
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0answers
16 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
3
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1answer
30 views

Reference request for stochastic process

I studied the book, "Probability with the book, Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal ...
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2answers
19 views

bayes theorem related problem. [on hold]

Suppose that Mr. Zafar becomes sick in the middle of the night and asks his sleepy wife to get some drug for him from the medicine cabinet. Two kinds of tablets are available T1 and T2. There are only ...
6
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0answers
89 views

Show two random variables have same distribution

Let X, Y be two non-negative random variables satisfying the condition $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. How can one show that X and Y are equal in ...
3
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3answers
38 views

You are making cookies and add N chips to dough randomly, and split it into 100 equal cookies, again at random. How many chips should go into dough?

Question: You are making chocolate chip cookies. You add N chips randomly to the dough and you randomly split the dough into 100 equal cookies. How many chips should go into the dough to give a ...
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0answers
18 views

Proof $Var(X|Y)=E(X^2|Y)-E^2(X|Y)$ [on hold]

Is it true and if yes, then how to prove it (please do not use the law of total variance because I need this relationship to prove the law itself): $$Var(X|Y)=E(X^2|Y)-E^2(X|Y)$$
3
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1answer
30 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
0
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2answers
25 views

Finding the probability of a complex event…

My apologies - I'm not a math professional, but the guys in my office (a bunch of web programmers) just came across across a logic problem that we've been discussing. We have a solution, but now ...
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0answers
12 views

probability theory and application btech 2nd year mike [on hold]

A class consisting of 4 graduate and 12 undergraduate students is randomly divided into 4 groups of 4. What is the probability that each group includes a graduate student?
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1answer
28 views

Probability of exactly $K$ events out of possible $N$

So I've stumbled upon this question in Grimmett and Stirzaker's text. I have their solutions manual, which starts off like this: The line above, where the statement is expanded into sums, is where ...
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2answers
36 views

Exponential distribution of random variable [on hold]

Random variable $X$ has probability density function $g(x)=\frac{3}{7}x^2\mathbf{1}_{[1,2]}$. Is there a function $F: \mathbb{R}\to\mathbb{R}$ for which $F(X)$ has an exponential distribution with ...
0
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0answers
23 views

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. [on hold]

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. I want to calculate the probability that the random vector $(X, Y )$ lies in the set $A$. ...
-1
votes
1answer
26 views

Men and Women enter a supermarket according to independent poisson process (stochastic process) [on hold]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
1
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1answer
30 views

Transitivity of a stochastic order

Let $X$, $Y$, $Z$ be three independent random variables such that $P(X \geq Y) \geq 1/2$, $P(Y \geq Z) \geq 1/2$. Is it true that $P(X \geq Z) \geq 1/2$? It seems true but I'm having a hard time with ...
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2answers
55 views

Expected values of a dice game with a 30-sided die and a 20-sided die.

Two people, $A$ and $B$, have a $30$-sided and $20$-sided die, respectively. Each rolls their die, and the person with the highest roll wins. ($B$ also wins in the event of a tie.) The loser pays ...
0
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1answer
49 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
2
votes
3answers
412 views

Product of two infinite sequences

Let $p_i$ be reals in (0,1) such that $\sum_1^{\infty} p_i=\infty$ and $\sum_1^{\infty} (1-p_i)=\infty$. Prove that $\sum_1^{\infty} p_i(1-p_i)=\infty$. I know a probabilistic proof (follows from ...
1
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1answer
22 views

The hierarchy of Liapounov conditions

The general setup is an array $(X_{nj} : n \in \mathbb{N}, 1 \leq j \leq k_n)$ of random variables (where of course each $k_n$ is an integer of value at least $1$). Write $S_{n} := \sum_{j=1}^{k_n} ...
0
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0answers
40 views

Find probability of complex event.

Okay, so today I've had task in my test, which is supposed to be hard: Cafe serves donuts. Every day number of students eat in this cafe. $\frac{3}{5}$ of these students are engineers, and ...
2
votes
1answer
55 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
2
votes
1answer
32 views

Computer Component with Gamma Distribution?

I comes to a question of one old-exam as follows: if the life of one computer component (in year) has Gamma Distribution (if I translate correctly) with ...