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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687 views

Covariance of two random variable, with one uniformly distributed and the other dependent on it.

Problem from actuarial Exam P: Let $X$ and $Y$ denote the values of two stocks at the end of a five-year period. $X$ is uniformly distributed on the interval $(0,12)$. Given $X=x$, $Y$ is ...
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0answers
212 views

Distributing cards among players

Moderator Note: This is a current contest question on codechef.com. N players sit around a round table. There are $n \cdot m$ cards with unique numbers of range $1\ldots n\cdot m$. Each player ...
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2answers
237 views

Expected number of coin tosses needed until all coins show heads

We flip $n$ fair coins every iteration of the game. Every coin that shows heads is removed from the game and we use the remaining $n-k$ coins to play the game again (where $k$ is the number of heads ...
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1answer
39 views

How to show that the following limit converges to some order of 1/n term .

How to prove that $$[e^{t\sqrt{\frac{1-p}{np}}}-1-t\cdot\sqrt{\frac{1-p}{np}}-\frac{1}{2}t^2(\frac{1-p}{np})]\cdot p ...
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2answers
41 views

What is the probability when i attempt twice?

// I have never got probabilities' lessons , I only know some basics Let's say the probability of me hitting the target ( let's consider hitting a bottle with a soccer ball ) in ONE ATTEMPT is ...
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1answer
100 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
2
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1answer
93 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
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1answer
26 views

Expected number of games played with variable trigger and retrigger of extra games

I've attempted to solve this problem, but I'm not 100% sure I'm correct. Problem: Consider a game played with the following extra game trigger conditions: There is a 0.001 probability of triggering ...
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1answer
289 views

Probability question on uniform distribution

I need help with the following question: A computer in adding numbers rounds each number to its nearest integer. Suppose that all rounding errors are independent and uniformly distributed over ...
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3answers
171 views

What is expected number of cards drawn when a card is chosen from a 52 deck and game ends on picking a heart?

I need help going about solving this problem: Suppose a card game is played using a standard 52 card deck. Drawing a heart ends the game. Drawing anything other than a heart continues the game. ...
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3answers
218 views

Find the probability that the $4$th ball removed from the box is white

A box has $10$ balls, $6$ of which are black and $4$ of which are white. $3$ balls are removed from the box, their color unnoted. Find the probability that a fourth ball removed from the box is ...
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2answers
130 views

Pick coloured balls from given urns

The contents of three given urns I, II and III are as follows 1 white, 2 black and 3 red 2 white, 1 black and 1 red 4 white, 5 black and 3 red One urn is chosen at random and two balls are drawn. ...
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1answer
267 views

2D random walk variation

If a point on a 2D lattice is allowed to take a random walk by taking a unit step either up, down, left or right, there is probability $1$ of reaching any point (including the starting point) as the ...
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1answer
81 views

Is the characteristic function of a multivariate normal distribution a real analytic function?

The characteristic function of a multivariate normal distribution with mean $\mu \in \mathbb R^n$ and covariance $\Sigma \in \mathbb R^{n \times n}$ is given by \begin{align*} e^{it^T\mu - ...
2
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0answers
72 views

Probability of getting not two head consecutively on tossing a coin ten times?

I am working on this problem and found total number of favourable cases 144 as $$1 + 10 + \binom{9}{2}+ \binom{8}{3}+ \binom{7}{4}+ \binom{6}{5}$$ and answer in the book is 128 favourable cases ...
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1answer
56 views

Prove that a Modified Cantor Distribution is Atomic.

Consider a measurable space $\{\mathcal{I},\mathcal{B}\}$, where $\mathcal{I} = [0,1]$ and $\mathcal{B}$ are the Borel sets on $\mathcal{I}$. And also, denote $\mathcal{C}$ as the cantor set on ...
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4answers
144 views

probability of getting two jokers out of 999 cards

There are 999 Cards with two jokers. three person(Say a,b,c) draw cards so each will have 333 cards at the end. what is the probability for only one person getting 2 jokers at the end? (doesn't ...
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1answer
697 views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
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2answers
1k views

SAT Probability of 4…

X _ _ X The figure above represents four offices that will be assigned randomly to four employees, one employee per office. If Karen and Tina are two of the four employees, what is the probability ...
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1answer
47 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
1
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1answer
85 views

Need help with a basic exercise about Markov chains

Suppose $\left\{ X_{n}\right\} _{n=1}^{\infty}$ is a Markov Chain taking real values. Are the following Markov Chains? $$Y_{n}=\sum_{i=1}^{n}X_{i} , Z_{n}=\left(X_{n},X_{n-1}\right)$$ Edit1 I ...
3
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4answers
732 views

Probability of Winning a Contest

This is my first question so apologies if its unclear/vague. There exists a contest with me in it, and $5$ others, thus $6$ people in total, along with $5$ prizes. A person can only win one prize, ...
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1answer
53 views

Predict number of Birthdays for 1000 person of same class in next 365 Days

I want to know an approximate number of birthdays for a class where each month 1000 Persons are added up. Like 1st month its 1000, 2nd month its 2000, 3rd month it is 3000 And so on. Now lets say ...
3
votes
3answers
86 views

convergence to exponential with order 1/n

We know that limit $\left(1+\dfrac{x}{n}\right)^n$ converges to $e^x$ but how can we prove that limit $\left(1+\dfrac{x}{n}+o(\frac{1}{n})\right)^n$ converges to $e^x$.
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1answer
49 views

Distribution of marbles on number line

I have a set of marbles and a number line from 0 to infinity. Every step I either put a new marble on the number 0 or I move one existing marble (chosen uniformly) to the next number. The ratio ...
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2answers
103 views

Find probability in card question

Suppose that a test for extrasensory perception consists of naming (in any order) $3$ cards randomly drawn from a deck of $13$ cards. Find the probability that by chance alone, the person will ...
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1answer
24 views

Find the probability of selecting an ordered pair from set $S$

Define $S=\left\{ \left( a,b\right) \in \mathbb{N}\times \mathbb{N} \mid a\leq 10,b\leq 10\right\}$ , randomly choose an ordered pair from set $S$. Find the probability that makes $\dfrac ...
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0answers
49 views

Prove the duality law for events

There is this law in the book I'm reading, but I can't prove it, neither can I find a proof online. It's 'called the duality law, and it's as follows : If $A$, $B$, and $C$ are events then: $AC + BC ...
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1answer
43 views

Statistical independence test, is it real?

Statistical independence does not mean real independence. In a roll of die experiment let $A=\{2,4\}$. $P(A) = \frac{1}{3}$. Let $B = \{2,1,3\}$. $P(B) = \frac{1}{2}$. $P(A|B) = \frac{1}{3} = P(A)$. ...
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5answers
2k views

What does the decomposition, weak union and contraction rule mean for conditional probability and what are their proofs?

I was reading Koller's book on Probabilistic Graphical Models and was wondering what the decomposition, weak union and contraction properties of conditional probability mean. But before I ask exactly ...
3
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1answer
117 views

Show that convergence in the mean implies convergence of the means [closed]

Question: Let $X_n$, n = 1,... denote a sequence of real-valued random variables; $X_n$ is said to converge in mean if $\hspace{20mm}$$$\lim_{n\to\infty} E[|X_n-X|] = 0$$ Show that if $X_n$ ...
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1answer
52 views

combinatorics & probability problem

There are numbered cards 1 to 13 each of colour red, green, yellow and white. And four players have been distributed 4 each of these cards randomly. What is the probability that each player gets ...
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1answer
44 views

calculating moments of random variables

Suppose that $E[X]$ is known where $X$ is a random variable. Is it possible to compute $E[X^3]$ with only this information? I am thinking of resorting to moment generating functions but I am not sure ...
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1answer
43 views

A moment's question.

Let G be a (absolutely) continuous distribution such that $$\displaystyle{\int_{-\infty}^{\infty}{x^{2}dG(x)}}<\infty$$ or $$\displaystyle{\int_{0}^{1}{\left[G^{-}(t)\right]^{2}dt}}<\infty.$$ ...
29
votes
1answer
614 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
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0answers
93 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
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1answer
26 views

Creating a bivariate distribution from two independent variables

If you have two random variables that are independent say $X\sim f_X (vars)$ and $Y \sim f_Y (vars)$. Is this a way to produce a bivariate distribution $f_{(X,Y)}$? $f_{(X,Y)} = p(X=x \cap Y=y) = ...
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1answer
50 views

Iterate through n coins flipping these obtaining all possible combinations.

If I have let say n coins all facing the same way. Is there an iterative method for turning these coins, one at a time, until all possible combinations have occurred one and only one time? This is ...
5
votes
1answer
256 views

What is the expected value of the number of randomly chosen real numbers between $0$ and $1$ needed to reach a sum of $1$? [duplicate]

My friend told me that the answer to this question was $e$, which intrigued me, but he refused to tell me why. My initial intuition was completely wrong. I thought that since the expected value of ...
1
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1answer
23 views

Application of Bayes' Thm.

I know that for this problem you would use Bayes' Theorem, but I am having issues figuring out which pieces would be of value. So far I have: P(cancer) = .008 P(accurate test given cancer) = .95 ...
1
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1answer
72 views

Looking for peculiar vector transformation

I have a vector of numbers from 0 to 1. For example: [0.5, 0.5, 0.1]. I need to find a transformation which increases sum of the vector to asked number and: -keeps the order of elements (if element1 ...
1
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1answer
78 views

What's the distribution of $\Phi(X), X \in N(0,1)$?

In a course on statistics, this set of non-compulsory exercises were supplied (in Swedish). I'm stuck on 8.10. My translation of the exercise: The stochastic variable X has a $N(0,1)$ ...
12
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7answers
520 views

The probability that x birthdays lie within n days of each other

This is a question that has bugged me for quite some time: what is the chance that x people happen to have their birthdays within n days of each other? A bit more specific, since this is how a ...
1
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1answer
408 views

given lognormal distribution, find expected value of its function

I know that $Y(t)$ is a lognormal function with $$E[\log(Y(t)]=\log(Y_0)-10t$$ and $$Var(\log(Y(t))=2t$$ Given this information, how do I find $$E[(Y(t)+3)^2]?$$ I'm guessing I need to somehow use ...
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1answer
19 views

Further Conditioning upon already Conditional Expectation

Let's say that $Y$ as a sample space of $\{1,2\}$ and $Z$ has a sample space of $\{3,4\}$. I know that $E[X]=E[X|Y=1]P(Y=1)+E[X|Y=1]P(Y=2).$ Now suppose I now want to further condition upon $Z$. I ...
1
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1answer
45 views

simple probability question - Independent experiments

p=0.43 (1-p)=q=0.57 n=3 Its known that the first experiment failed. what is the probability that the next experiments will succeed. I'm thinking P(X='Fail then success then ...
3
votes
1answer
114 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
1
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1answer
93 views

Conditional independence $a\bot b|\emptyset$ for $p(a,b,c) = p(a)p(c|a)p(b|c)$

Suppose a graph which factorizes: $p(a,b,c) = p(a)p(c|a)p(b|c)$ How can I show that $p(a,b) = p(a) \sum_c p(c|a) p(b|c) = p(a)p(b|a)$ as it is not shown in my textbook
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0answers
35 views

Showing $\lambda_V(x)\leq \min\{\lambda_1(x),\cdots, \lambda_n(x)\}$.

Suppose $X_1, \cdots, X_n$ are independent, nonnegative continuous functions, each $X_i$ has hazard function $\lambda_i(x)$. If $V=\max\{X_1, \cdots, X_n\}$, I need to show that ...
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0answers
46 views

For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...