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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11 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: ...
0
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0answers
11 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
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1answer
13 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 ...
2
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1answer
24 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
13 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 ...
0
votes
2answers
15 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
31 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 ...
0
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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
votes
2answers
21 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
25 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: ...
0
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0answers
14 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 $\hat\theta _{ML}(x_1,\ldots ,x_ n)$. ...
1
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2answers
272 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
47 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
votes
1answer
33 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
votes
1answer
29 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
votes
3answers
31 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 ...
-1
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0answers
24 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 ...
1
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0answers
21 views

Consider a random walk where $p \neq 1/2$, where the starting point is random and has a binom distn. Find the probability of absorption at $N$.

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, ...
1
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0answers
43 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 ...
1
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1answer
33 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: ...
1
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1answer
28 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 ...
1
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0answers
16 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
19 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: ...
0
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0answers
20 views

Combination of historical and current data in statistics

I have a general question about a statistical matter. Lets assume there exists a true and unique probability $p$ such that an event $X$ happens in the next 12 months. There is some information about ...
0
votes
1answer
26 views

What is the probability that on a given day, the number of half gallon containers provided is enough?

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. ...
0
votes
1answer
40 views

Check My Work on a Poisson Process/Distribution Question

I'm just curious if my work is correct, and if not, where I made a mistake. My Task: Cars arrive according to a Poisson process with a rate of 12 per hour. (1) What is the probability that the ...
0
votes
1answer
26 views

Standard deviation: calculating how polarizing a question is

I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer. Answer a: $30\%$ Answer b: ...
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0answers
36 views

Four points inside a rectangle [on hold]

We randomly choose 4 points inside a rectangle.What is the probability that they lie in the same half ?
2
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3answers
33 views

2 restaurants located randomly

any help on following question will be much appreciated. Q. Suppose that $2$ restaurants are going to be located at a street that is $10$ km long. The location of each restaurant is chosen randomly. ...
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0answers
22 views

Probability of a train journey

A trip from south east London to Southampton consists of three journeys: bus journey to Crystal Palace station, train journey from Crystal Palace to Clapham Junction, train journey from Clapham ...
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0answers
20 views

Easy Question from Application: Estimate for transition probabilities of random walk - finding a coupling

SHORT VERSION: Find appropriate Coupling Suppose we have a random walk on the natural numbers, where we go to the left with probability $p_L \geq \frac{1}{6}$, to the right with probability $p_R\leq ...
3
votes
2answers
46 views

What is the intuition of why convergence in distribution does not imply convergence in probability

For me its very counter intuitive (that convergence in Probability and Distribution are not the same), because, conceptually, if two random variables have the same distribution, then they should be ...
0
votes
1answer
15 views

Question about assigning probabilities to elementary events

Let $(\Omega,S)$ be a sample space with a probability function $P$. Then, the book by Rohartgi that I am reading says that: if $\Omega$ is uncountable, one cannot assign positive probability to ...
2
votes
2answers
42 views

9 room probability and expected value

I got the following question: In a house with 9 rooms. There is 1 mouse that is looking for some food. This can be found in 2 rooms, but there are also 2 cats, these are in different rooms. When the ...
2
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3answers
25 views

Horse racing question probability

Been thinking about this for a while. Horse Campaign length: 10 starts Horse Runs this campaign: 5 Horse will is guaranteed to win 1 in 10 this campaign Question: what is the Probability of ...
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0answers
24 views

Challenging probability and statistics problems?

The books I have on the subject lack entertaining problems. Can someone suggest a book with more challenging problems - perhaps not at olympiad level, but a bit easier. Thanks very much!
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0answers
25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
0
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2answers
53 views

Radon-Nikodym derivative of Measures [on hold]

Im having some trouble reconciling what I thought I learned about RN Derivatives as they relate to probability measures wikipedia,lecture notes with this blog post by John Baez mentioning it as it ...
2
votes
2answers
35 views

Binomial distribution central moment calculation

If for a binomial distribution the mean is $4$ and variance is $3$, find th $3^{\text{rd}}$ central moment. I understand that the first and second central moments are mean and variance ...
4
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4answers
103 views

Casino turns 50% of your losses into “free play”, are odds in your favor?

As a limited-time promotion, if you gamble during your first week at this casino, and you suffer a net loss of money, the casino will give you half of your losses (up to a certain amount) as "free ...
1
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1answer
21 views

Poisson process problem

Problem statement below... Customers of a store frequent it at a rate of 0.75 customers per minute following a Poisson process. Exactly one customer goes into the store during a 2 minute span. ...
0
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1answer
13 views

Exercise on iid sequence of uniformly distributed random variables (and LLN).

I'm trying to solve following problem: Let $X_{1}, Y_{1}, X_{2}, Y_{2},\ldots$ - iid, from uniform distribution on $[0,1]$, $f\colon[0,1]\rightarrow[0,1]$ be measurable and $Z_{j} = ...
1
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1answer
30 views

gaming - How to calculate odds of roulette “Strategy”

I know that this strategy shouldn't work, but I can't seem to get the math to work to make it fail, and when I model it; it succeeds. I'm obviously missing something, but can't see what. In American ...
0
votes
1answer
27 views

Same card probability in a deck

Working on my first board game design. To simplify, having 4 types of cards: 5 blue ones, 4 red ones, 3 yellow ones, 2 green ones, so 14 cards in total. How can I calulate the possibility that ...
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1answer
19 views

How to show that the 3 events are independent but not pairwise independent [on hold]

Say the experiment is tossing 2 dice and the events are A - first die is a even number B - sum of both dice is 4 C - outcome of the two dice differ by at most by 2 What I got is $(A \cap B ...
1
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1answer
38 views

I can't find my mistake in this gambler's ruin problem.

I am trying to solve a problem in a game. In this game there is a card that says something like spend 1 resource and flip a coin. If the result is heads, you receive 2 of the resource back, if tails, ...