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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27 views

Interesting and challenging problem

I've been given this problem to solve, but didn't succeed until now. Can you help me? A city has 5 billion paper money (bills) in circulation. Thirty million of them are taken daily to the bank ...
1
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2answers
13 views

Probability of choosing two bulbs with the same rating given that one has a specific rating

I am trying to teach myself statistics, and working through Jay DeVore's excellent text of "Probability and Statistics for Engineering and the Sciences". The problem is as follows: We have box of the ...
0
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1answer
8 views

Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
0
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0answers
5 views

probability generating function moments for the multivariate case

Suppose ${\bf X} = (X_1, \ldots, X_d)$ is a non-negative integer-valued random vector, with pmf $p$, I tried to extend the results of the univariate generating function to the multivariate case, is ...
0
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2answers
27 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
1
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1answer
17 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
0
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0answers
3 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
1
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1answer
24 views

probability of 26 letters

A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random. A. What is the probanility that the word HAMLET appears somewhere in the string if letters? ...
0
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1answer
15 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
0
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1answer
33 views

A measure theory question-1

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
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1answer
24 views

Probability of last cheese

I hope that someone could help me with understanding the exercise. In a cycle shaped house there are n chambers. In this house there is a mouse and each chamber has cheese except the room where the ...
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2answers
24 views

A probability theory question

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
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3answers
46 views

Intuition behind independence result

The following problem is from Wasserman's $\textit{All Of Statistic}s$. I have worked through the algebra to arrive at the result, but it still seems very strange to me, so I would appreciate any ...
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0answers
4 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
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0answers
15 views

Expected value - product of functions of uniformly distributed variables

We have $n$ random variables $X_1,...,X_n$, $n\geq 2$, where $X_i∼U(0,1)$ and all of them are iid. Let $ Z=\min(X_1,...,X_n)$ and $ \overline{X} = \frac{1}{n}\sum_{i=1}^{n}{X_i}$. Calculate ...
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0answers
10 views

Invariant distribution of a Markov chain

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

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
5
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1answer
16 views

simulating a fair six with a four equal sector spinner

Whist teaching basic probability I needed a group to use a fair four sector spinner but I'd none left. I gave them a die asking them to disregard 5,6 should they arise. The problem got me thinking ...
0
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1answer
32 views

proving a statement based on probability theory [on hold]

Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$
3
votes
1answer
39 views

Why this solution of the birthday problem is wrong? [duplicate]

If we have $n$ people there are $n(n-1)/2$ possible pairs that we can find. The probability that any two people have the same birthday is $1/365$. So for $n$ people the probability of finding at least ...
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3answers
18 views

Given the percentage, what's the probability it will happen exactly?

If a drug is effective $75\%$ of the time, what's the probability that it will be effective on EXACTLY $15$ out of $20$ people. Is there a formula or list of steps for this type of question?
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2answers
20 views

Can't find intersection of two probabilities.

I have the following problem: While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation ...
0
votes
1answer
11 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 ...
2
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0answers
23 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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2answers
26 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 ...
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2answers
34 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 $$ ...
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0answers
34 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 $Z=Y/X,$ ...
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0answers
21 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 ...
0
votes
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 $t_1, t_2\in\mathbb{R}$ ...
1
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1answer
20 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 = ...
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0answers
16 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 ...
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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. ...
1
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2answers
28 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, ...
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0answers
15 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
15 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 $X_n \overset{P}{\rightarrow} 0 \implies X_n \overset{{L^P} }{\rightarrow}0$
0
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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 : ...
0
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1answer
17 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 ...
1
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2answers
57 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} ...
3
votes
2answers
62 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 ...
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). ...
1
vote
1answer
21 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
17 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 ...
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0answers
34 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 ...
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0answers
12 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 ...
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votes
1answer
29 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 ...
0
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1answer
56 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
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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 ...
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0answers
9 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 ...
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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 ...
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1answer
34 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 ...