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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7 views

Conditional expected value not mutually indipendent sets

Let be $E,G,H$ pairwise independent events but not mutual (e.g. $\mathbb{P}(E\cap H)=\mathbb{P}(E)\mathbb{P}(H),\,\mathbb{P}(G\cap H)=\mathbb{P}(G)\mathbb{P}(H), ...but \,\mathbb{P}(E\cap G\cap H)\ne\...
1
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0answers
12 views

Maximize sum with no two consecutive variables

Random variables $x_1,x_2,\dots,x_{100}$ are drawn independently from the uniform distribution over $(0,1)$. After knowing the values, we are allowed to choose a subset of them as long as no two ...
-1
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2answers
32 views

What is the probability of random walking ant to be at a position after some finite steps on an infinite grid?

Is it even calculable? What if the grid is infinitely dimensional? Lets say that it is a simple random walk, and probability to move to any neighboring position is equal, but other types are also ...
5
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3answers
124 views

Is there a quick way to justify that this elementary probability is equal to $2/3$?

I just solved this problem with the conditional probability formula and after a while the answer was surprisingly $2/3$. I believe there must be a tricky short way to calculate it. Can somebody help ...
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3answers
29 views

Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
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2answers
27 views

Why does a null-hypothesis have to have a definite value?

In hypothesis testing, why does the null hypothesis (H_0) have to have one defined value?
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2answers
37 views

Vaiance of the sum of correlated variables

If the variance of two correlated variables is: $$Var(r_1+r_2)=\sigma^2_1+\sigma^2_2+2\textrm{cov}(r_1,r_2)=\sigma^2_1+\sigma^2_2+2\rho\sigma_1\sigma_2$$ where $r_1$ and $r_2$ are vectors, then what ...
2
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1answer
45 views

What is the Probability came from the same machine

Machine A produced 65 of the day’s output of Product X and machine B produced the other 55. If three products are selected with replacement at random from the day’s output, the probability that, My ...
1
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1answer
164 views

How does maximum distance from the ecliptic determine the frequency for which a planet is likely to be occulted by the Moon?

Consider that Venus can stray ~7 degrees from the ecliptic and that Jupiter only strays a maximum of ~1.8 degrees from the ecliptic. The Moon strays up to about 5 degrees from the ecliptic. When the ...
8
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1answer
45 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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0answers
139 views
+50

Discover to which batch a coin belongs

The following question is taken from a book, in a chapter on probabilty: You have two batches of unbalanced coins. One has coins which turn up head with probability $p_1$, and the other has coins ...
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0answers
15 views

Differences of Markov chain is Markov

In my studies of Markov chains, I was tackled with this tough problem: Let $ \{ X_n \}_{n=0}^{\infty} $ be a homogeneous Markov chain with transition probabilities satisfying $ | i-j | > 1 \to ...
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0answers
11 views

Forecasting for a non-steady state problem

Let say, at each discrete time step, $t_i$, we can forecast of a specific event's occurring rate, $I_{trans}$ by following formulation: $$I_{trans}(t) = I_{ss}. \sqrt{(\frac{A\tau}{t})} .e^{-\frac{B\...
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1answer
31 views
2
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1answer
30 views

Arrange 18 pips on a die with at least one 0 side to maximize the probability that 5 rolls sum to 13 or more.

You are arranging pips on a standard 6-sided dice. Rules: At least one side must be left blank at 0. The average roll must be 3 (so, you have 18 pips to distribute among five sides). You want to ...
3
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0answers
21 views

Monotone Class Theorem and another similar theorem.

I found different statements of the Monotone Class Theorem. On probability Essentials (Jean Jacod and Philip Protter) the Monotone Class Theorem (Theorem 6.2, page 36) is stated as follows: Let $\...
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2answers
22 views

Estimating a random variable from repeated trials

I have an $n$ sided die and suspect that it is biased. I'm interested in the probability of rolling a $1$, so I roll the die $m$ times and count up the number of times I roll $1$, then divide the ...
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0answers
6 views

Drift analysis of an absorbing Markov chain

Consider a set $S$, and suppose we have a sequence of random subsets $$ \zeta_t = \{x_1, \dots, x_n\} $$ for $x_1, \dots, x_n \in S$. We do not know with which probability density the points of each $\...
0
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1answer
80 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
6
votes
3answers
81 views

Probability of choosing $n$ numbers from $\{1, \dots, 2n\}$ so that $n$ is 3rd in size

We uniformly randomly choose $n$ numbers out of $2n$ numbers from the group $\{1, \dots, 2n\}$ so that order matters and repetitions are allowed. What is the probability that $n$ is the $3^{\text{rd}}$...
4
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7answers
539 views

Puzzle about technique of fair using of unfair coin

There is an unfair coin. It tends to land on one side more than on the other. It is unknown which side is it. There is Mr. A and Mr. B. They argue about something and they want to use that coin to ...
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1answer
50 views

How to choose between two options with a biased coin

We would like to choose between theatre and cinema by tossing a coin. Unfortunately the only available coin we have has probapility of heads $p\ \left(\dfrac{1}{2}<p<1\right)$. How could we use ...
2
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2answers
51 views

If we've got 10 coupons, what is expected number of different ones if there are 25 different types

I can't figure out this problem : There are 25 different types of coupon, all equally probable to get. If we have got 10 coupons, what is expected number of different coupons between them? ...
7
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1answer
689 views

Intuition on Harris recurrence

I am trying to get some intuition on Harris recurrence in Markov chains. Define state space $\mathcal S$ comprising a single communication class, $f_{ii}^{(n)}=P(X_n=i, X_{n-1}\ne i,\ldots X_1\ne i\...
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1answer
579 views

Statistics question Conditional Probability

Question: Of three cards, one is painted red on both sides; one is painted black on both sides; and one is painted red on one side and black on the other. A card is randomly chosen and placed on a ...
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0answers
30 views

Probabilistic parameters determination

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
5
votes
1answer
45 views

Difficulty understanding step in Kac's proof of Feynman-Kac Theorem

I am trying to understand a proof of the Feynman-Kac Theorem, as set out in Mark Kac's 1949 paper 'On Distributions of Certain Wiener Functionals'. Kac defines a series of independent and ...
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2answers
47 views

Find the probabilty of 25 random people

X is the weight of one person, $X \sim N(\mu =78,\sigma =13.15 )$. If I choose randomly 25 people, what is the probability that the average of their weights will be $86$ ? I define $\displaystyle Z =...
0
votes
1answer
26 views

Conditional probability by joint probability

I have the joint pdf $$f(x,y)=\frac{1}{5}(3xy^2+2x^3y);0<x<1;0<y<2$$ and I have to calculate $$P(\frac{1}{2}<Y|X<\frac{1}{2})$$ I have found that $$f_{X}(x)=\int_{0}^{2}\frac{1}{5}(...
0
votes
1answer
27 views

Are $X_1$ and $X_2$ independent?

Let $X=(X_1,X_2)$ be an absolute continues random vector with the density function $f_X(x_1,x_2) = \left\{ \begin{array}{ll} \frac{2}{3}x_1+\frac{4}{3}x_1 x_2+\frac{2}{3}x_2, & \mbox{for } (...
1
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1answer
39 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
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3answers
54 views

True or false:if $A\subset B$, then $P(A)<P(B)$?

They ask me if this statement is true or false, and explain why. They suggest I write an example showing why it is false or true. The statement is: if $A\subset B$, then $P(A)<P(B)$. What I ...
0
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0answers
20 views

Bounding Probability Distribution

I have the following problem. Let $X$ be a continuous random variable with image $[0, b]$ for some finite $b>0$. So we have finite moments, $\mathbb{E}[X^n]$. I am hoping to say something about the ...
2
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1answer
46 views

Which has higher variance, coin toss vs dice roll?

Dusting off some high school stats and getting confused over the following: Two betting games: Pick right side of coin, even-money bet ($p = 0.5$, $q= 0.5$), Pick right value in a 10-sided ...
3
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1answer
34 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
0
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0answers
13 views

Is there any example of a Markov chain (discrete) with limit distribution (discrete) of heavy tail?

Is there any example of a Markov chain with limit distribution (discrete) of heavy tail? In other words, a Markov chain whose limit distribution has infinite second moment?Already, thanks for the help!...
5
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1answer
1k views

Fingerprint match probability

I am trying to use the formula for the birthday paradox as a reference to figure out an equation that represents the probability of a fingerprint match. Here's the equation for probability of a ...
1
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1answer
667 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
1
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0answers
35 views

looking for a probability function which satisfies the following conditions

I am looking for a continuous probability function of$f(a,p,x)$ which satisfies the following conditions $a$ is a positive constant $0 \le p \le 1$ is a positive constant $x > 0$ is the variable $...
2
votes
2answers
74 views

I choose three random integer point in origin $|x|, |y|\leq r$. plane, what probability to this point creates a right triangle?

I wont to choose three random integer point in origin $|x|\leq r, |y|\leq r$ at plane as $(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})$. What the probability that this three point create a right triangle ...
0
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0answers
28 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
2
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1answer
339 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
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0answers
17 views

Data transmission process PDF

Given the quasi-defined data transmission random process: $X(t) =\sum_{n=-\infty}^{+\infty} a_n \pi_T(t - nT)$ where $a_n$ are statistically independent RVs that can either assume the value 0 or 1 ...
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0answers
12 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
2
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1answer
16 views

The intuition behind conditional probability and independence in the case of different sample space

I came up with this question when doing this problem: In throwing a pair of dice, let A be the event that "the first die turns up odd", B the event that "the second die turns up odd", and C the ...
0
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1answer
28 views

Probability of a permutation for inversions

Sample space for following problem is S4. And the probability $p(\sigma)$ of a permutation is $\alpha$ times the number of inversions of $\sigma$ for suitable $\alpha$. We have to find the value of $\...
2
votes
2answers
27 views

References for the applications of probability in gambling

The intuition behind many theorems in probability comes from gamblers' games. I would like to know if there are any books or articles which cover some such connections between probability and its ...
-1
votes
2answers
62 views

Expected number of tosses to get 3 consecutive Heads [on hold]

I have a fair coin. What is the expected number of tosses to get three Heads in a row? I have looked at similar past questions such as Expected Number of Coin Tosses to Get Five Consecutive Heads ...