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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1answer
13 views

Measurablity of functions defined over sections of product measures

I have to solve the following exercise but I am unable to proceed. Could you please give me some hints to how to solve it? Let $(\Omega_1, \mathcal{F}_1)$ and $(\Omega_2, \mathcal{F}_2)$ be ...
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2answers
21 views

Distribtution of the maximum of three uniform random variables.

How do I get the cumulative density function of $Y$? $X$ is a continuous random variable with pdf $$f(x) = 1,\quad 0 < x < 1. $$ Three independent observations of $X$ are made. Find the pdf ...
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0answers
23 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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0answers
11 views

Is the expected value of a monotone function on a uniformly distributed random variable monotone?

Consider the following definition: A sequence of uniformly distributed random variables $(X_n)_{n \in \mathbb n}$ where $X_{n-1} \sim U[a_{n-1},b_{n-1}]$ and $X_n \sim U[a_{n},b_{n}]$ such that $a_n ...
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0answers
6 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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2answers
34 views

How to find the expected value of the radius of the meat ball? Assuming its shape is a perfect sphere? [on hold]

To make a meatball, you choose beef with probability $2/3$ and turkey with probability $1/3$. If you choose beef, the number of ounces you take is uniformly distributed between $2$ and $4$. If you ...
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1answer
17 views

Probability of rolling higher than $N$ by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides.

I'd like help finding a formula for the probability of rolling higher than a target number, $N$, by summing the highest $X$ number of dice out of a set $Y$ number of dice, each with $Z$ sides, ...
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0answers
15 views

conditional probability, logical product

I was working my way through Kruschke's textbook and got to Chapter 9 and the result on factoring out conditional probabilities for hierarchical models, seemed similar to something in Feller Vol1 ...
2
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2answers
17 views

The covariance between $X$ and $Y$.

Suppose that $X$ and $Y$ are both continuous random variables that have a joint probability density that is uniform over the rectangle given by the four $(x,y)$ coordinates $(0,0)$ , $(2.46,0)$ , ...
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1answer
27 views

Suppose there are 5 dollar bills in a box…find the PMF

Suppose that there are $5$ dollar bills in a box: three $1$ dollar bills, one $5$ dollar bill and one $10$ dollar bill. You are allowed to pick up two bills at the same time from the box randomly. Let ...
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0answers
14 views

Deriving sample size using Hoeffding's Inequality

I want to use Hoeffding's Inequality to determine the necessary sample size $n$ to construct a confidence interval of $\epsilon$ and $\alpha$. I've consulted the Wikipedia article and am confused as ...
1
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1answer
19 views

Breaking sides of equation to prove a probability.

I am trying to prove that $$P(B \cap C \mid A) = P(B \mid A) P(C \mid A \cap B)$$ So far I have been trying to break the equation down LHS and RHS, but I am having trouble figuring out the right ...
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0answers
6 views

renewal function and constant

Let F a density and $U(t)=\sum_{n=0}^{\infty}F^{*n}(t)$ with $F^{*n}(t)=F*...*F$ Show if for all $a<\infty$, $F(a)<\infty$ then for all $t<\infty$ and $\delta<1$, there exist ...
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0answers
11 views

Distribution function derivative bounds give bounds on associated measures? Billingsley theorem 31.4 proof.

I am working through Billingsley, Probability & Measure. Struggling with the proof of theorem 31.4: Suppose $u(a,b) = F(b) - F(a)$ and that $F'$ exists throughout a Borel set $A$. If $F' ≤ c$ ...
1
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1answer
29 views

The sum of all combinations greater than $x$

Suppose I choose $3$ integers at random from $\{1,…, 100\}$. What is the chance that the sum of those integers exceeds some number $x$? I know the probability that the numbers will sum to a ...
1
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1answer
31 views

Function of a markov chain $f(x)=x^3$

I have given a Markov Chain $X_n$ with the state space $\{0,1,2\}$ and the transition Matrix $$P= \begin{Bmatrix} 0.3 & 0.2 & 0.5 \\ 0.5 & 0 & 0.5 \\ 0.2 & 0.1 & 0.7 ...
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0answers
24 views

Probability of passing an exam with random % [on hold]

A student gives an exam with 4 questions, 1 right answer.He knows the answer for 20% , he knows that 30% has 2 wrong answers,and at 50% he don't know anything. What's the probability to answer at 1 ...
2
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2answers
31 views

Throw a fair die repeatedly until I have thrown three sixes

I play a game in which I have to throw a fair die repeatedly until I have thrown three sixes, after which I stop and note the total number of throws. What is the probability that I take six throws? I ...
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0answers
14 views

jumps and equality(exercice)

Let $N_t$ be a renewal process and $T_n$ the jumps with $T_n=X_1+...+X_n$.$X_1,..,X_n$ where $X_i$'s are independent random variables identically distributed law $F_X$ and $T_{-1}=T_0=0$ Let ...
0
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1answer
17 views

How many possible outcomes of getting 2 face values when drawing 6 cards from a deck of 52?

We draw six cards from a deck of 52 playing cards. How many possible outcomes of getting 2 face values? I don't understand what it means when it says "how many possible outcomes of getting 2 face ...
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0answers
13 views

What is the likelyhood that they will see all the 5 posters? in5 different elevators? [on hold]

We are putting up 5 different posters in 5 different elevators for 5 days. there is 300 staff accessing the elevators. the elevators cover 6 floors. Staff normally go up to their office 4 times a day ...
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0answers
10 views

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
27 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
32 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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1answer
25 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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2answers
44 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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0answers
14 views

Entropy of sum of uniform random variables on a simplex [duplicate]

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 ...
0
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1answer
38 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
19 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 ...
0
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1answer
35 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
14 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 ...
0
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1answer
20 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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0answers
25 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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0answers
42 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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0answers
37 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 ...
0
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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 ...
0
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1answer
29 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 ...
2
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4answers
73 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 ...
2
votes
1answer
22 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 ...
0
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1answer
30 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: ...
2
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0answers
19 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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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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3answers
28 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
52 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 ...
1
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1answer
12 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 ...
0
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2answers
28 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 ...
1
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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
22 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| ...
0
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1answer
17 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 ...