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
22 views

Integration limits of a Marginal Probability Density Function with a Triangle-Shaped Boundary

I have given a triangle shaped boundary $M$ of my probability density function in $\mathrm{R}^{2}$, with the limitations beeing: $$y = 0$$ $$y = x$$ $$y = 2-x$$ and a probability density function $$ ...
0
votes
1answer
5 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
0
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0answers
10 views

Is this Markov chain irreducible?

Let $(X_n)_{n \in \mathbb{N}}$ be a Markov chain with statespace $I = \{0,1\}^m$ and transition probabilities $$p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 & ...
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2answers
39 views

Probability in a Restaurant

In a revolving restaurant, there are four round tables each with three seats. How many different ways can $12$ people sit in this restaurant? This is what I think the answer is: $$\binom{12}{4} ...
0
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0answers
23 views
+50

Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:- A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
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2answers
22 views

Determing a transition probability matrix

I need some support with this homework exercise: An urn contains at most $N$ balls. Let $X_n$ be the number of balls in the urn after the $n$-th execution of the following procedure: If the urn is not ...
2
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0answers
13 views

Probabilistic Logic

I was wondering if there is any system of logic that has been worked out that explicitly uses probabilistic notions at its foundation. It would include ideas like as a first principle, all statements ...
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0answers
22 views

Find the density of a ratio of random variables

$X$ has density $2x, 0 < x < 1,$ and $Y$ has density $1/10$ over $0 < y < 10$. $X$ and $Y$ are independent. I have to find (a) density of $Y/X$ (b) $E[Y/X]$ (c) $E[Y^2/X]$ I let $Z=Y/X,$ ...
3
votes
2answers
443 views

Poisson Distribution when only given using mean

I'm doing the following homework problem and am unsure of whether or not my answers are correct. This is my first time working with Poisson distribution and I want to make sure I am doing it ...
1
vote
1answer
18 views

probability density and distribution functions

I have $6$ independent and identically distributed variables such that $C_i \sim N(1000,400)$. 1) Calculate the density functions, distribution function and characteristic function of $C = ...
-1
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0answers
10 views

Confidence interval of exponential random variables

I have a sequence of random variables $X_1, X_2, ..., X_n$ such that $X_i = e^{-(x_i-Θ)}$ I have to construct a confidence interval of the form $[Θ−c,Θ]$,where $Θ = \min _i{X_i}$. For $n = 10$ how ...
0
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1answer
16 views

independence and characteristic functions [duplicate]

Why is it that \begin{equation*} \mathbf{E} [e^{i t_1 X_1} e^{i t_2 X_2}] =\mathbf{E} [e^{i t_1 X_1}]\mathbf{E} [e^{i t_2 X_2}] \end{equation*} for RVs $X_1, X_2$ and all $t_1, t_2\in\mathbb{R}$ ...
0
votes
0answers
18 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
-1
votes
0answers
107 views

Pareto distribution,fisher information, confidence interval

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
0
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0answers
7 views

Galton Watson process

Let $X_n$ the number of individus of the $n^{th}$ generation. For example suppose that a father has no brother and sister and does $3$ children. Suppose that thefather is the generation $0$ (i.e. ...
0
votes
1answer
1k views

Probability(A union B complement) given P(A) = .15, P(B) = .10, P(A intersect B) = .04

This is a homework question that I'm stuck on and I'm looking to see if I'm going about it the right way and how to put the pieces together. So far I know that I can get $P(A \cup B)$ which is $0.15 + ...
0
votes
1answer
12 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
1
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2answers
24 views

What's the chance of $(\frac{1}{2})^x$ with $y$ iterations?

If I have a program that creates, let's say, one billion integers, with each having a pure $50 - 50$ chance to be one or zero, what is the chance of finding $x$ zeros in a row? for brownie points, ...
5
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0answers
41 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 ...
-2
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0answers
13 views

Theoretical probability that everyone in the U.S. is separated by 6 degrees [on hold]

The six-degrees-of-separation theory says that I can be most certain that I have a friend who has a friend, who has a friend, who has a friend, who has a friend, who has a friend, who is friends with ...
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0answers
13 views

Convercenge in probability implies convergence in Lp [on hold]

Show that if $X_n$ is that $|X_n|< C$, with $C\in \mathbb{R}$, $\forall n \in \mathbb{N}$, then $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
3
votes
2answers
59 views

What are the odds of any role of a 24 sided die occurring 4 or more times in 10 rolls?

Note that I am not asking about the odds of a chosen roll happening 4 times in 10 rolls, (this has a probability of 0.000517 according to a binomial calculator), rather, the odds of ANY roll happening ...
2
votes
1answer
855 views

Conditional independence property: weak union

Let (X,Y,W,Z) be disjoint sets of random variables each with finite space. Then prove that if $Pr(X|W,Y \cup Z)=Pr(X|W)$ then $Pr(X|Y,Z \cup W)=Pr(X|Z \cup W)$. This is sometimes referred to as weak ...
1
vote
1answer
18 views

Hypergeometric function variance

In a fishing event, a small lake is populated with $75$ trout, among which $25$ are tagged. Each participant is allowed to capture $5$ fish during the day (the fish are not put back into the lake). ...
0
votes
1answer
15 views

Confidence Interval Question - Steps Taken, no given standard deviation

I just wanted to make sure I was doing this Confidence Interval problem correctly (or incorrectly). Q: The following are the daily number of steps taken by a certain individual in 20 weekdays. (some ...
3
votes
1answer
332 views

Looking for first course textbooks on probability and statistics for math majors

I am taking a probability and statistics course soon and would like to find a text book that is targeted more towards math majors rather than engineers (which is what this class is). The book my ...
0
votes
2answers
37 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 ...
0
votes
0answers
11 views

How to recompute the markov transition matrix given a reduction to the number of states? Clustering from a transistion matrix

I am been puzzled with this one for sometime. Given a transition matrix (as below) for a markov chain of N states; how do we calculate the transition matrix for N-1 states, where we combined stat n1 ...
1
vote
1answer
20 views

Two related question, in one. Same topic: Dispersion..

$1.$ Prove: If $X_1,X_2,X_3,\ldots,X_n$ are independent random variables then: $$D\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n D(X_i)$$ Proof: Because of independence we have: $$D(\sum_{i=1}^n ...
0
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0answers
16 views

Proving that each element in reservoir have equal probability of been selected in reservoir sampling?

Here is the description of the algorithm and proof of the correctness The algorithm creates a "reservoir" array of size $k$ and populates it with the first $k$ items of $S$. It then iterates through ...
0
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0answers
24 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
-1
votes
1answer
27 views

What is the probability of picking two black cards out of a pack of ten?

I have ten cards; eight of them are red, and the remaining two are black. What is the probability of choosing both black cards in four draws? I have tried $\frac{3 \cdot 4}{2} \cdot \frac{3 \cdot 3 ...
4
votes
4answers
174 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 ...
0
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0answers
48 views

combinatorics contest problem

Question: Calvin has a bag containing $50$ red balls, $50$ blue balls, and $30$ yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out $5$ more red ...
0
votes
2answers
30 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. ...
1
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5answers
2k views

Coin chosen is two headed coin in this probability question

I have a probability question that reads: Question: A box has three coins. One has two heads, another two tails and the last is a fair coin. A coin is chosen at random, and comes up head. What ...
0
votes
1answer
19 views

A die is thrown $n$ times. $X_1$-number of times a number from $\{1,2,3\}$…

.. $X_2$ number of numbers that fell from $\{4,5\}$, $X_3$ number of $6's$ that fell. Find $$P\{ X_1=k\mid X_2=m\};0\leq m \leq n.$$ Now, I believe that $X_3$ is completely irrelevant here. What I ...
0
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0answers
8 views

Probability in Entity Linking

This question is about computer science probability, in particular Natural Language Processing, but I think that there is a little too much math in order to ask it on stackoverflow. Anyway, I'm ...
0
votes
2answers
21 views

In a box which has balls numbered 1..100 , 5 balls are drawn.

$X$- random variable that represents the largest number of the 5 drawn. Find the distribution of $X$. Now, it seems that this random variable is of discrete type. What I have trouble it defining it ...
3
votes
1answer
30 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 ...
3
votes
3answers
30 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 ...
0
votes
1answer
292 views

I am trying to solve probability that 2 events do not occur

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the ...
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3answers
34 views

probability about coins [on hold]

A gambler has two coins in his pocket, a fair coin and a two-headed coin. He picks one at random from his pocket, flips it and gets heads. What is the probability that he flipped the fair coin?
1
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2answers
25 views

Distances to the center of points uniformly distributed in a disk

We choose $n$ points at random from the surface of disk of radius $1$ (the points are chosen with equal probability). If we omit the point furthest from the center (from $n$ points), what is the ...
0
votes
1answer
33 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
1
vote
1answer
296 views

Gradient-descent and Hidden Markov Models

I would like to use gradient-descent to fit the parameters of a simple 2-state HMM. This paper Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983), An Introduction to the Application of the ...
0
votes
1answer
23 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 ...
1
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0answers
25 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
1
vote
1answer
15 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
1
vote
1answer
79 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 ...