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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8 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, ...
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0answers
26 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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0answers
22 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 ...
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1answer
29 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
26 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
45 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
23 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
12 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
14 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
44 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: ...
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0answers
19 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 ...
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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
21 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
30 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
21 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
18 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 ...
2
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2answers
30 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
14 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
38 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
24 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
21 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
51 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
32 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
102 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
votes
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
26 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, ...
-1
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0answers
45 views

Expectation of an interval

Given $g(\theta) := Pr\{X\leq\theta\leq Y\}$ with $Y\geq X$, what is $E[Z]$ where $Z:= Y-X$ ? Also $X{\not\perp}Y$ Progress: $$X\leq\theta\leq Y\Rightarrow \{Z \geq \theta-X\}\cap \{Z\geq\ ...
1
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1answer
16 views

Computing expectation of a function of two random variables

I have two arrays $X$ and $Y$ of length $N$ each. In array $X$, I have random numbers $x_1$, $x_2,\ldots,x_N$, whose sum is $S_x$. Similarly in array $Y$, I have random numbers $y_1$, ...
1
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0answers
13 views

Probability of Detection confidence interval

I am dealing a Probability of Density curve. I have the curve fitted but need to apply a $95\%$ confidence interval for that curve. The parameters of the curve are as follows: $$-5.0793 x^2+ 4.853x- ...
1
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1answer
28 views

The probability of getting at least 5 balls of the same color from a uniformly distributed set of size 126 with red, blue and white balls

I am trying to calculate the following problem: I have a uniformly distributed set consisting of red, blue and white balls. This set contains a total of 126 balls, thus there are 42 (${126 \over ...
1
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1answer
22 views

Density function of minimum of random variables

Let $\{ X_i \}_{i=1, \dots,n }$ a set of i.i.d random variables whose density is defined by $f(\theta,x)=e^{-(x-\theta)}$ for $x>\theta$ and $f(\theta,x)=0$ for $x<\theta$. Where $\theta$ is a ...
0
votes
1answer
22 views

Law of a random variable

I have a probability space $(\Omega,F,P)$ with $\Omega=R,F=B(R),P(A)=\int_A xe^{-x}1_{x>0} dx$ for $A \in F$. I have a function $X:\Omega\rightarrow R$, $X(\omega)=\omega$. If ...
1
vote
1answer
25 views

Probability of dominating set in random balanced tournament

I'm trying to estimate some probability in a random tournament, and I know that what I have is false, as it leads to contradicting results published some 40 years ago. But I don't know where the ...
0
votes
1answer
49 views

What is A intersect B

Two dice are rolled A - The first die shows an even number B - The sum of the two die is 4 Now the question is for A it states only the first die so the second dice roll shouldn't factor in ...
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0answers
25 views

Measurable maps in metric spaces.

i have several questions about measurability of maps with values in metric spaces : 1/ When $X$ and $Y$ are two separable metric spaces, it is easy to prove that $\mathcal{B}(X\times Y) = ...
6
votes
2answers
58 views

what is the probability that the circumcircle of 3 point

Mary picks any three non-collinear points inside a given circle, what is the probability that the circumcircle of these 3 points will be covered by the original circle? This is from a test ...
0
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0answers
44 views

Mega-straight flush with a huger hand

Three days ago I asked about the probability of drawing a straight flush when being dealt $26$ out of the $52$ cards of the deck, which Michael wisely solved. Now I'd like to make things more ...
0
votes
1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
0
votes
1answer
42 views

A CD players plays songs until a certain artist is selected.

Suppose you have a CD player with 10 CDs where 9 of those CDs belong to an artist A and 1 CD belongs to an artist B. Suppose this CD player will pick a CD at random, and from that CD it will pick ...
0
votes
1answer
26 views

Probability generating function of binomial distribution [duplicate]

In a population of $2n$ individuals there are $n$ infected individuals and $n$ uninfected. Suppose that $X$ of the n uninfected become infected, where $X \sim \mathcal B(n, p)$, and, then, given $X = ...
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0answers
13 views

Skorokhod vs Meyer zheng topology

I am new to the Skorokhod space and I want to know why Meyer-Zheng topology on the space of càdàg functions is weaker than the standard Skorokhod topology. Thanks in advance!
0
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0answers
23 views

Maximizes the expected utility, Decisition making, statistics

QUESTION Can someone help me figuring out how to calculate this question, i just learned this stuff and i haven't do any example like this before... So i'm interpreted this question as: We need ...
0
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0answers
19 views

Sum of Gaussian and Binomial distribution

I need to calculate the probability of sum of two probability variable, each of which is distributed as binomial distribution and Gaussian respectively. I mean how to calculate the probability of ...