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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Need help with probability homework

Alright so I could use some help with my homework, thank you in advance! Plura goes to the gym 15% of the days of the year. Carla goes to the gym 20% of the days of the year. a) what is the ...
2
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1answer
21 views

Expectation problems in probability.

Dan tosses a coin $n$ times independently, while the probability for a unique tail is $1\over 3$. For $1\le k\le n$, let us denote the number of sub-sequence of length $k$ of H's. For example, if n=5 ...
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2answers
22 views

During a night, each chameleon changes its colour to one of the other four colours with equal probability.

Five chameleons of all different colours meet one evening. During the night, each chameleon changes its colour to one of the other four colours with equal probability. Find the probability that the ...
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2answers
18 views

Counting problem: ways of opening stores in non-adjacent blocks?

A coffee company wants to set up stores along the main street of town, which has $n$ blocks. The company won’t open two stores in the same block, or in two adjacent blocks. Q: For this coffee shop, ...
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3answers
33 views

how to solve this conditional probability

Manufacture A and B produce one type of electrical element, given that the probability of produced element being faulty is $0.05$ for A and $0.01$ for B. If two of these elements has been picked, from ...
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0answers
15 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
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1answer
21 views

Probability of co-occurence

Of total $N$ people, $m$ people are good at mathematics and $c$ people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...
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0answers
12 views

proof of the convergence of confidence intervals

The confidence interval can be derived intuitively by replacing the standardized mean with the standard normal and variance with sample variance, but is there a formal limit? I'm trying to prove if ...
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3answers
154 views

Find the probability that the final score is 4 in a dice game with two throws

A game uses an unbiased die with faces numbered 1 to 6. The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then die is thrown again and the ...
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0answers
32 views

Probabilities: Meeting people

The probability of women meeting a man is $m$. Let's look at the perspective of a specific man. The probability of meeting him is $\tilde m$. Say women look twice for men. Then (assuming $\tilde m$ ...
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1answer
16 views

Convergence of random variables under different probability measures

I have a succession of random variables $X_n$ on $\Omega=[0,1]$ with $X_n=(1-\omega)^n$. I have to prove the convergence almost sure and/or in law in these case: $\mathbb P=\delta_{0}$ $\mathbb ...
2
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0answers
25 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
-1
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1answer
20 views

Derivative of a CDF [on hold]

Suppose that $X$ is a random variable whose mean is $m$. I need to show that $\frac{\partial}{\partial m} \text{Prob}\{X\geq x\} >0$. Intuitively, increasing the mean I'm shifting probability ...
2
votes
3answers
21 views

Property of cumulative distribution function

I was taking the course on random variables , where I faced below property of cumulative distribution function: $$\lim_{x\rightarrow a^+}F_X(x)=F_X(a^+)=F_X(a)\qquad\qquad ...
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0answers
21 views

Prove: $E[E(X|X+Y))=E(X) ; E[X|X+Y] = n/(n+m)*(X+Y)$ [duplicate]

E[E(X|X+Y)]=E(X) is diferent of E[X|X+Y]. And in E[X|X+Y] I give the final result that is "n/(n+m)(X+Y)", and I am asking to demonstrate that E[X|X+Y] = n/(n+m)(X+Y). Let X and Y be independent and ...
1
vote
1answer
26 views

Probability distribution for a geometric distribution don't add up to 1

Say I'm rolling 2 dies,numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore,probability of success=0.25 and prob of failure=0.75. This is an example of a geometric ...
3
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0answers
28 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
0
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1answer
16 views

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2… $M_N$.

In a box there are $M_1$ balls numbered 1, $M_2$ numbered 2... $M_N$. From the box $n$ balls are drawn without returns. Find the mathematical expectation of the number of numbers that are not drawn. ...
0
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1answer
26 views

In a box there are 3 white, two black balls. Players A,B,C one by one draw balls from the box. Find the probability of winning each player.

White balls are returned to the box, black balls are kept. A player has won when he has drawn the last ball(black). And if $X$ is the random variable the represents the number of draws , find the ...
4
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2answers
43 views

Probability when cutting the stick twice

Given a stick of length $l$. We cut the stick twice. Let $X$ be the random variable defined by the length of the stick after the first cut, and $Y$ be the random variable defined by the length of ...
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1answer
24 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
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1answer
24 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
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0answers
28 views

Mastermind Probability Distribution [on hold]

I was thinking of the game MasterMind, a game with 6 different colored pegs and a 4 part code using those colors. There can be more than one peg of the same color in the code. For example ...
1
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1answer
39 views

Calculating complicated expectation

I need to calculate $\operatorname{E}( X_2 \mid X_1=x, Y=y)$, where $Y=\max\{X_2,X_3\}$ and joint density of $X_1$, $X_2$ and $X_3$ is given by: ...
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0answers
23 views

Is the almost surely limit of measurable functions measurable in probability spaces?

Suppose we have $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{F}_n$ a sub $\sigma$-algebra of $\mathcal{F}$. Let $(X_n)_{n=1}^\infty$ be a sequence of $\mathcal{F}_n$-measurable functions converging ...
2
votes
1answer
15 views

Box with balls of different colours. Probability of finding a specific colour.

A box has $10$ red balls and $5$ black balls. A ball is selected from the box. If the ball is red, it is returned to the box. If the ball is black, it and $2$ additional black balls are added to the ...
3
votes
1answer
55 views

Random Variables and Moment Generating functions

Let $(X_i)_{i∈\Bbb{N^+}}$ be a sequence of i.i.d random variables and for $n ∈ \Bbb{N^+}$ set $S_n := \sum _{i=1}^{n} X_i$ and $Y_n := max(X_1, . . . , X_n)$. Assume that the moment generating ...
1
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1answer
14 views

Convergence in probability

I need to prove that given the r.v. Xn with the same distribution functions, the sequence of r.v. Xn/n tends to 0 in probability. Following the definition i find: P(|Xn/n| > a) = P(|Xn| > na) for ...
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votes
2answers
18 views

CDF of a Uniform probability density function

I want to find Cumulative distribution function (CDF) of the following density function: $ f(x)= \begin{cases} 3/20 & \text{for } 2 \leq x \leq 4 \\[8pt] 4/20 & \text{for }4 < x \leq ...
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3answers
40 views

Probability of choosing two numbers so they differ by at least 2

A box has $10$ balls numbered $1,2, \dots, 10.$ A ball is picked at random and then a second ball is picked at random from the remaining nine balls. Find the probability that the numbers on the two ...
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1answer
26 views

If mutiplication of probabilities of two events is equal to their intersection,then are the events always independent?

Here is an example , Let a ball be drawn from an urn containing four balls, numbered $1, 2, 3, 4$. Let $E = \{1, 2\}$, $F = \{1, 3\}$ If all four outcomes are assumed equally likely,then we have ...
2
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2answers
24 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
0
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2answers
28 views

Conditional expectation of random variable

I have this home assignment in Introduction to Probability, and I'm not comfortable with definitions and heuristics. I really need someone to check if I'm even in the right direction. The question: ...
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0answers
16 views

Maximum likelihood estimator and confidence interval

Let $\theta$ be an unknown constant. Let $W_1,…,W_n$ be independent exponential random variables each with parameter $1$. Let $X_i=θ+W_i$. First, I need to find $\hat\theta _{ML}(x_1,\ldots ,x_ n)$. ...
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2answers
280 views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
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2answers
50 views

Chances of this… [on hold]

9 people sat in a circle. They wrote their name on a piece of paper, folded it over and placed it in a hat. The hat was shuffled to mix up the pieces of paper. The first person picked out the name ...
3
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1answer
40 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
3
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1answer
30 views

probability question that just seems to easy to be the case

the game of mastermind starts in the following way: one player selects four pegs, each having six possible colors, places them in a line. the second player then tries to guess the sequence of colors. ...
0
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3answers
32 views

Probability of a Rare Event Occurring within a Certain Number of Times

I'd like to know how to find the probability of an event occurring, given the probability of that event, within a certain number of chances for it to occur. For example, say that once every year ...
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0answers
29 views

Convergence in law and probability

I have a succession of random variables $\{X_n\}$ with $P(X_n=3)=1/n^2$ and $P(X_n=4)=1-1/n^2$. It's defined $Y_n=nX_n$ and i have to prove the convergence almost surely,in law and in probability. I ...
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0answers
24 views

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
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0answers
62 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
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1answer
34 views

“Inverse” of nondecreasing, right-continuous function?

Suppose $F : \mathbb{R} \to \mathbb{R}$ is a nondecreasing and right-continuous function. Define $G : [\inf F,\sup F] \to \overline{\mathbb{R}}$ by $G(p)=\inf \{ x : F(x) \geq p \}$, with the ...
1
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1answer
31 views

Independence of Random Variables

If $X$ and $Y$ are independent random variables so are also the random variables $f(X)$ and $g(Y)$ for $f$ and $g$ measurable and bounded functions. The independence of $X$ and $Y$ implies: ...
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1answer
29 views

Multiple examination of a result (probability)

A performs a task and submits the result to B and C for examination. B confirms the result. C thinks the result is wrong. The reliability of A is 0.7, for B is 0.8 and that of C is 0.9. (reliablity = ...
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2answers
49 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
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0answers
24 views

probability of getting lucky in exam? [on hold]

In an examination, you are given a choice to pick up a chit, which has a question, there are ten of those chits(randomly arranged), only half you have prepared(you know all the question but you're ...
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0answers
17 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
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0answers
22 views

Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional $\phi$ on the Wasserstein ...
0
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1answer
46 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...