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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Stumped - How would I solve this probability question?

This question was merely a fun online math problem to see how many people could solve it, but I haven't been able to since last week and it's begiing to drive me nuts. The question: A man has 7 math ...
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7 views

Distribution of the maximum of correlated Gaussian variables

I try to unterstand how the distribution of the maximum of a iid Gaussian sample changes when we introduce correlation. For this, I simulated iid random samples and correlated samples with $\sigma =1$ ...
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2answers
15 views

Probability of X given the sum

I am given that $X \sim P(\lambda)$, $Y \sim P(\gamma)$, and told to calculate the distribution of $P(X | X+Y = n)$ I proceed as follows $$ \begin{equation} \begin{split} P(X=i|X+Y=n) &= ...
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2answers
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How to calculate $V(X+Y)$ with $X$,$Y$ dependent?

I want to calculate the covariance of two dependent variables $X$ and $Y$ and I don't know the value of $V(X+Y)$, that is, the variance of $X+Y$. I know how the quantities relate to each other: ...
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11 views

nearest point in Poisson point process

Consider a homogeneous Poisson point process with density $\lambda$. Consider a sector with an arbitrary origin $o$, angle $0<\theta<2\pi$. What is the PDF of distance from the origin to $n$-th ...
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Question concerning a proof on Stroock and Varadhan 1971

In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads: where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ ...
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1answer
17 views

prior probablity is lower than conditional probabilty

Is there any reason behind the fact that prior probability is always lower than the conditional probability? I'm wondering why this happens..Can you please provide any reason for it?
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1answer
11 views

Expectation of size of bootstrapped sample

Lets say we have a sample $\mathbf{X} = \{x_1, x_2, \dots, x_N\}$. We draw $N$ points from $\mathbf{X}$ with replacement (do a $\textit{bootstrap})$. What is the expectation of size of bootstrapped ...
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3answers
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Probability Of People Visiting Pubs

5 people went out to pubs. everyone individually choose a pub randomly from 10 pubs available. What is the probability that at "Pub1" "Pub2" came at least one person? $|\Omega|=10^5$ How can ...
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2answers
30 views

A grasshopper starts at the origin and is equally likely to hop north,s,e,w. What is the probability that it's coordinates will be 0,0 after 4 hops?

The grasshopper must hop in all $4$ directions (North, South, East, and West) to get back to the origin after $4$ hops. Therefore, I did: $\frac{(4 \cdot 3 \cdot 2 \cdot1)}{4^4} = .09375$. However, ...
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0answers
12 views

Finding the factorial moment generating function

I need help finding $G_x(t)$ $f(x)= pq^{x-1}$ for x = 1, 2,... and 0 otherwise. I know $G_x(t)= M_x(ln t)$ I have started the following $$\sum_{x=1}^\infty e^{xlnt}f(x)$$ $$\sum_{x=1}^\infty ...
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Golf Question when you have uneven players teams how can you make it simple and fair

We have various sizes groups sometimes they are even sometimes not. Most of the time we have foursomes equal groups sometimes we might have 3 foursomes one threesomes. We play golf we come into the ...
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3answers
24 views

Probability picking colored stones that match a series on a card.

I'm inventing a board game that requires a person to randomly pick 4 colored gems out of a bag and have them match colored gems on a card. If I use ten each of four different colored gems, would ...
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2answers
20 views

How to calculate multiplication of probabilities when range is given?

Assume that probability of $A$ is $0.6$ and probability of $B$ is at least $0.75$. Then how do I calculate the probability of both $A$ and $B$ happening together?
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11 views

Probability of visiting each cell of an $N\times M$ grid

Consider an N by M grid. Rows are numbered $1$ to N, from top to bottom. Columns are numbered $1$ to M, from left to right. We are initially at cell $(1, 1)$ and want to go to cell (N, M). From any ...
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1answer
34 views

Arranging Couples

4 married couples entering a restaurant, there is only one table available, therefore the waiter put 4 people randomly near table and the 4 others near the bar, what is the probability that: ...
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7 views

Probabilistic Modelling of uncertain positions of objects in a 2D-Grid

I have a 2D-Grid which is populated by obstacles of different sizes. A size is always a whole number of cells. An obstacle is at least one cell big. If I did kown the size of the object but had only ...
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2answers
19 views

How to calculate the joint probability from two normal distributions

I have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A, B ...
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Ordering Books On a Shelf

The are 6 different finance books and 4 different math books, there were arranged randomly, what is the probability that books from the same subject are standing one by the other. The answer is ...
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0answers
7 views

Simpler proof of Karp's Theorem for probabilistic recurrence relations?

A probabilistic recurrence relation is of the form $T(x) = a(x) + T(h(x))$ with $a(x)$ deterministic (usually $a(x) = 1$) and $h(x)$ being a random variable over $[0,x]$, so that $T(x)$ itself is a ...
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How to reduce the standard deviation from $P$ where $P$ involves a random integer uniformly distributed in $\left[ 0,100\right]$

I have a probability $P$ derived from: - A random integer $A$ uniformly distributed on its range such that $A\in\left[0, 100\right]$ - An integer $K$ such that $K\in\Bbb N$ - A number $X$ such that ...
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2answers
236 views

Probability- Guessing names of people

In the newspaper "Pop World," an old photo of well-known pop singers was published from when they were kids. The names of the singers are known, and one needs to identify the singers. If the ...
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Condition on Poisson random measure

It is well-known that given a $\sigma$-finite measure on a measurable space, there exists a Poisson random measure. I approach some proofs, but I want to know that if the condition of separable is ...
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1answer
17 views

Characteristic function of a non-negative random variable?

Is it possible to decide if a random variable is non-negative almost surely, by looking at the characteristic function of the random variable?
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18 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
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1answer
26 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow ...
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1answer
65 views

Expected value when die is rolled $N$ times

Suppose we have a die with $K$ faces with numbers from 1 to $K$ written on it, and integers $L$ and $F$ ($0 < L \leq K$). We roll it $N$ times. Let $a_i$ be the number of times (out of the $N$ ...
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34 views

What are the main kinds of mathematics? [on hold]

I stumble upon as much on math I don't know (trascendal math, number theory) and math I know on the internet and elsewhere. I have a pretty good idea about differential and integral calculus, and I'd ...
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1answer
16 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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1answer
19 views

Proof of equivalent probabilities in anomaly detection

In A New Look at Anomaly Detection there is a claim for the proof of probabilistic definition of normal is as follows, a guess of the probability for event i is $\pi_i$, the true probability is ...
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1answer
14 views

sum of discrete and absolutely continuous random variables

We know a sum of two independent absolutely continuous random variables is still absolutely continuous since its density function is the convolution. And without being independent, the sum may be ...
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Are all orthogonal projections conditional expectations?

When will orthogonal projections coincide with conditional expectations? Does that have something to do with the fact that not all closed subspace are probability spaces? Is it why when we fix a ...
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1answer
18 views

permutations vs combinations on slot machines with repeating elements on each reel

For a slot machine with 5 reels where there are repeated elements on each of the reel. Example: Reel 1 [ 1, 1, 2, 1, 3, 5, 6 ] Reel 2 [ 1, 2, 3, 4, 5, 5 ] Reel 3 [ 2, 2, 3, 2, 4 ] Reel 4 [ 1, 2, 3, ...
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1answer
46 views

Density of probability in a square [on hold]

Suppose we have a square $$\{(x,y) : x \in [0,1], y \in [0,1] \}.$$ We suppose that we have $X$ and $Y$ are the coordinates in this square that are uniformly distributed. Why does the joint density is ...
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1answer
19 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
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1answer
25 views

Probability with conditional involved

Alan is part of a dinghy sailing team. The probability of strong winds tomorrow is 0.3. In the strong winds the probability of Alan's team winning falls to 0.25. Calculate the probability that Alan's ...
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1answer
47 views

Probability a blackjack dealer will bust if you know their score and know the exact deck?

If you know the exact cards left in a deck, and the score of the dealer, how can you calculate the exact probability that they will bust? The dealer behaves as follows: If the dealer's score is less ...
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3answers
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How much would you pay to pull a ball from the bag?

If there are 9 white balls and one black ball in a bag. The white balls are valued at 10 dollars and black ball at 100 dollars. How much are you willing to pay for each pull from the bag (only one ...
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1answer
22 views

How to construct a transition matrix?

I'm giving my first steps in stochastic processes but I'm having some difficulties. See the following example Suppose that whether or not it rains today depends on previous weather conditions ...
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2answers
52 views

Why birthday distribution is not uniform. [on hold]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...
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0answers
17 views

Probability of a decay process [on hold]

Consider a decay source with decay constant $\lambda$ (exponential decay) arriving with rate $R$ (poisson process). What is the probability that the product comes from the source right before it? (A ...
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3answers
155 views

How to understand independence of probability?

By definition, when $$P(E\,|\,F) = P(E)$$ holds, we say that $E$ is independent of $F$. By definition of conditional probability, $$P(E\,|\,F) = {P(E \cap F) \over P(F)} \Rightarrow P(E \cap F) = ...
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2answers
107 views

Game of probability

n a game, played between $2$ players there is a circular field and one of the players is blindfolded, who stands in the center of the field. The other player stands at a fixed point on the ...
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2answers
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Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
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finding the limit for a martingale [duplicate]

I have trouble finding the exact limit for a martingale: Let $\{\xi_n\}_{n\in\mathbb{N}}$ be a Markov chain with $\xi_0$ uniformly distributed in $[0,1]$ and $$ ...
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2answers
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Calculate dependent probabilities [on hold]

Imagine that two raffles will happen, every raffle will reward 1 person. 10 people will participate. The first raffle will reward 1 person. The second raffle will reward another person, but the ...
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0answers
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Bayes theorem with multiple conditions

how to calculate P(a|b,c,d). knowing that b, c and d are NOT independent from each others ? i know how to solve it if there is independency assumption. however, i am just wondering if there is any ...
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1answer
22 views

Show that $|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$

Consider the random variables $X$ and $Y$ defined in the same space $\Omega$. Show that $$|F_{X,Y}(x,y)|^2\leq F_X(x)F_Y(y)$$ This question comes from an old test, I know that ...
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2answers
31 views

Find the density of their average

If $f_{X,Y,Z}(x,y,z)=e^{-(x+y+z)}I_{[0,\infty]}(x)I_{[0,\infty]}(y)I_{[0,\infty]}(z)$ find the density of their average $\frac{X+Y+Z}{3}$ I'm a little lost on how to solve this exercise, ...