This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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Bounding entries of random vector

Given a random vector $\mathbf{e} \in \mathbb{R}^n$, is it possible to count (or bound) the number of entries in $\mathbf{e}$ that have $|e_i| \ge 1/ \sqrt{n}$? It is known that entries in ...
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1answer
29 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
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1answer
17 views

Ehrenfest chain

In the Ehrenfest model, let $X_n$ denotes the number of balls in the left urn. And there are $N$ balls total. When we calculate $P(X_{n+1}=i+1|X_n=i, X_{n-1}=i_{n-1},...,X_0=i_0)$, why don't we take ...
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0answers
15 views

What are the recent real life use or applications of the Cauchy Random Variable?

We have a short assignment on the described question and I already have gone through a lot of trash results from Google. I can't seem to find any. I don't know where else to post this question. ...
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0answers
23 views

Probability that a given function is prime…

If we have a set of primes $p_1$, $p_2$, ... , $p_n$, we can easily construct a function of their product: $$f(\alpha) = \alpha \left( \prod_{k=1}^n{p_k} \right) + 1, \alpha \in \mathbb{N}$$ I'm ...
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2answers
393 views

Upper bound on the entropy of a sum two random variables

Let $X$ be a random variable such that $|X| \leq A$ almost surely, for some $A > 0$. Let $Z$ be independent of $X$ such that $Z \sim {\cal N}(0, N)$. My question is: How large can the entropy ...
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2answers
37 views

Probability Question

Each day that i arrive the platform on the underground station on my way back home, there is probability $0.177$ that i have to wait more than $3$ minutes for a train to arrive. What is the ...
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1answer
21 views

Expected payment of a roll of dice with rerolls

We've got the following game: You roll two dices. You get paid equal to the number rolled. Additionally, if you roll doubles, you reroll (Same rules apply to that roll. That means there's not limit ...
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1answer
34 views

If you roll three dice, what is the probability of getting at least two number are same? [on hold]

If you roll three dice,What is the probability of getting at least two numbers the same?
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0answers
11 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
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0answers
8 views

Entropy of sum of uniform random variables on a simplex

For two i.i.d random variables $X$ and $Y$, which are uniformly distributed on the $n$-dimensional simplex $\Delta_n= \left\{(x_1,\ldots,x_n): x_i \geq 0, \sum_i x_i \leq 1 \right\}$, I want to find ...
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0answers
18 views

Probability of decline

this is probably a simple question. I want to figure the odds of a portfolio of assets declining in value in one year. The odds of decline for each of the assets individually is 30%. The ...
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1answer
30 views

How come this Poisson formula equals 1

In Poisson Random Variable: $$\sum_{x=1}^\infty \frac{e^{-\lambda}\lambda^{x-1}}{(x-1)!}=1$$ Why does this equal $1$? What property is this?
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0answers
19 views

How to solve system of equilibrium probability state equations

I have started studying markov chains where i have these statistical equilibrium probability state equations.These equations are solved for a particular case $s_1=4,a_1=5,s_2=2, a_2=1$ and a 15*15 ...
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1answer
18 views

Variance of Signum Function of Two Random Variables

Let $ X $ and $Y$ be two random variables with means $\mu_X$ and $\mu_Y$ respectively, as well as variances $\sigma_X$ and $\sigma_Y$ (all of which exist). I am interested in computing the following ...
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1answer
41 views

Expected number of red balls in urn

We toss balls into urns. Denote with $x$ the number of balls in an urn. And $x_r$ denotes the number of red balls. The share of red balls among the balls is denoted as $P$. We toss these balls into ...
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1answer
361 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
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29 views

Limit theorems in measure theory

From probability theory/measure theory we know set of theorems such as Monotone convergence, dominated convergence or conditions like uniform integrability which deals with the general question of ...
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1answer
27 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
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1answer
42 views

Poisson Process Network Sniffer Problem.

The problem is: Consider a traffic sniffer that observes packet arrivals into a link. Packets arrive according to a Poisson process with rate $\lambda$. If the sniffer sees no packet over a period of ...
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0answers
34 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...
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1answer
21 views

Independent or not Independent events.

The Sample space is $\{1,2,3,4,5,6\}$ with uniform distribution. Two numbers $i$ and $j$ in the set $\{1,2,3,4,5,6\}$ have been singled out. For each outcome $s$ in $S$ let $X(s)$ be the answer to ...
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1answer
19 views

Conditional expectation of a random walk given that it is positive

Let $\{\xi_k\}$ is a sequence of iid random variables with $E(\xi_1)=0$ and $E(\xi_1)^2=\sigma^2<\infty$. Define the random walk $Y_n=\sum_{k=1}^n \xi_k$. Is it necessarily true that the ...
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1answer
390 views

Binomial Distribution for defects

I'm stuck on the following problem: A batch of components has arrived at a distributor. The batch can be characterized only if the proportion of defective components is at most 0.10. ...
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4answers
69 views

Given that $6$ men and $6$ women are divided into pairs, what is the probability that none of the women will sit with a man?

I've generalized the question I was given here for simplicity: $6$ men and $6$ women are to be paired for a bus trip. If the pairings are done randomly, what's the probability that no women will end ...
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1answer
27 views

Which of the following can NOT be the possible value of $P(A \cup B)$?

Let $A, B$ be two events with $P(A) = 0.2$ and $P(B) = 0.4$. Then which of the following cannot be the possible value of $P(A \cup B)$? A) $0.3$ B) $0.4$ C) $0.5$ D) $0.6$ I understand ...
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2answers
56 views

Combination - Distribution of gifts.

Seven different type of gifts are to be distributed among $10$ children. Every kind of gift must be at least given to one child. Then, how many combinations do we have? Note:You have $A, A, A....$ ...
2
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1answer
53 views

Number of inversions

Compute the sum of the number of inversions that appear in the elements of $S_n$. In other words find the total number of inversions that the elements of $S_n$ have combined. I mean how can we ...
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2answers
31 views

Gambler's ruin (Deck of cards).

A deck of $52$ cards is shuffled, and the cards are turned up one at a time until the first $A$ appears. Show that: $$P(\text{next hand is Ace of spade}) = P(\text{next hand is 2 of club}) = ...
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1answer
28 views

Probability Distribution: Verification of my Thinking

More than anything, I just need someone to confirm for me that I'm on the right track. So I have a table that has some random variable $X$ which has a probability distribution table of: ...
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2answers
70 views

The derivatives of the logarithm of a moment generating function

Let $M_{X}(t)$ be an mgf of $X$. Show that the first derivative of $\ln M_{X}(t)$ at $t=0$ is $\mathbb{E}[X]$ and the second derivative of $\ln M_{X}(t)$ at $t=0$ is $\text{Var}[X]$ I'm not ...
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0answers
16 views

Bounds for the ratio of probabilities of Poisson-Binomial Distribution

There are $N$ Bernoulli trials, where $m$ trails have probability of success $p$ and $N-m$ trails have probability of success $q=1-p$. Assume $p>q$. The number of successes is a random variable ...
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0answers
33 views

Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
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3answers
23 views

How does one find the density of the $k$th ordered statistic?

Let $X_1,\ldots,X_n$ be $n$ iid random variables. Suppose they are arranged in increasing order $$X_{(1)}\leq\cdots\leq X_{(n)}$$ The first ordered statistic is always the minimum of the sample ...
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1answer
9 views

Independence: norm v.s. direction of a standard multivariate normal vector

Suppose that $v\sim N(0,\sigma^2 I_n)$ and with $||\cdot||$ denoting the Euclidean norm, define $$ u=v/||v||\quad\text{and}\quad w=||v||. $$ I've been told that $u$ and $w$ are independent and I see ...
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0answers
13 views

Probability of a permutation for inversions

Sample space for following problem is S4. And the probability $p(\sigma)$ of a permutation is $\alpha$ times the number of inversions of $\sigma$ for suitable $\alpha$. We have to find the value of ...
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2answers
24 views

Conditional Probability [on hold]

Each time a shopper purchases a tube of toothpaste, he chooses either brand $A$ or brand $B$. Suppose that for each purchase after the first, the probability is $1/3$ that he will choose the same ...
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1answer
17 views

Set algebra and expected value, this manipulation is correct?

Im doing a problem where I must evaluate the expected value of random variable $XY$, where $Y=M-X$. My question, this manipulation is correct? $$\Bbb E[XY]=\Bbb E[X\cap Y]=\Bbb E[X\cap (M\cap ...
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0answers
21 views

Are continuous processes almost surely bounded?

Is any process with continuous sample paths almost surely bounded on a finite horizon? If this is true, let $\{X_t\}_{t \in [0, T]}$ be such a process with continuous sample paths. Then we have $|X_t| ...
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1answer
18 views

estimating probabilty [on hold]

I would like some help with a probability problem. Given that $23.5$% are obese, $22.7$% of Americans are smokers, and $4.7$% are both obese and smokers. Estimate the probablity of 1) A person that ...
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1answer
96 views

Graduate probability: bounding the third moment [on hold]

Consider a real-valued random variable $X$ with $E[X^4]=1$. We know that $E[X^3]\leq 1$. If also $E[X]\leq 0$, find an constant $c<1$ such that $E[X^3] \leq c$ and find the smallest constant $c$ ...
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0answers
10 views

Simplex Algorithm (Exercise 3.11.33 in Grimmett and Stirzaker's Probability and Random Processes)

There are $n \choose m$ points ranked in order of merit with no matches. You seek to reach the best, $B$. If you are at the $j$th best, you step to any one of the $j - 1$ better points, with equal ...
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0answers
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Confusing Conditional Probability question 68 [on hold]

The four top tennis players in the world A, B, C, and D are invited to a special tournament where the winner gets one million dollars. In round one, Player A plays player D and player B plays player ...
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1answer
20 views

Waiting time for two independent poisson processes

Order of Events in Poisson Processes Assume that you have two independent Poisson process, $N_1(t)$ with rate $\lambda_1$ and $N_2(t)$ with rate $\lambda_2$. The probability that $n$ events occur ...
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0answers
25 views

markov chain: 2 state chain

I have a machine. It has two states, broken or working. If it is working, then it will be broken with probability $q=0.1$. If the machine is working, I will make \$1000 dollar a day. If it is broken, ...
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1answer
33 views

Poisson probability of an event A before event B

I'm trying to calculate the probability of two poisson processes events happening one before the other, with two different $\lambda$s. The way I see it, I can word it as the probability of event $A$ ...
5
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3answers
212 views
+50

Exploding dice in a dice pool

Say we role $n$ identical, fair dice, each with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides are numbered from $1$ to $d$ with no repeating number, ...
2
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1answer
33 views

Probability of a random Permutation [on hold]

Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last ...
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13 views

Expected random generation

Background: I've been reading about how Dota deals with its random generation. There's another question on Gaming.SE about this, but it doesn't give a formula, which is what I'm looking for. ...
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1answer
16 views

Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...