Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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How to calculate probability that a team will win

Let's say there are 10 teams: A-J. Each team always participate in each of the game. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be ...
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31 views

What is the probability that one person does not sit next to another.

5 people around a round table. (A,B,C,D,E) They are placed randomly with equal probabilities of being placed in a seat. What is the probability that A does not sit next to C.
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72 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
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1answer
65 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
2
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1answer
79 views

Separation of points in a Poisson point process

Suppose I have a Poisson point process $\mu$ in $\mathbb{R}^2$, with driving measure absolutely continuous with respect to Lebesgue measure. For any $\epsilon > 0$, I can choose a rectangle $R$ ...
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69 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
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1answer
14 views

Tail estimates for Binomial with constant mean

The Chernoff Bound gives a good tail estimate for a Binomial Distribution, but only if the mean goes to infinity. However, for a constant mean, Chernoff bound does not help at all. Is there some ...
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55 views

how to find out combination from following situation

i have three number 1 2 3 which will always be in this order {123}, i want to find out number of cases can be made, like {1},{2},{23},{13},{12},{123}{3},{}. but each number has two states like "a" ...
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69 views

On the conditional expectation.

I want to prove that: if $E[M_t\mid\mathcal{F}_s]=0$ where $\mathcal{F}_s$ is the filtration generated by a stochastic process X knowing that $E[M_t\prod_0^n h_i(X_{t_i})]=0$ for all $n\in N,\quad ...
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3answers
26 views

What is the probability that exactly one of the dice shows 0

3 Dice: 2 Green dice with faces labelled 0,0,1,1,2,2 1 red die with faces labelled 1,2,3,4,5,6 2 are chosen at random and then thrown. What is the probability that exactly one shows 0. So either ...
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267 views

2 of 3 dice are selected randomly and thrown. What is the probability that one of the dice shows 6

1 red die with faces labelled 1, 2, 3, 4, 5, 6. 2 green dice labelled 0, 0, 1, 1, 2, 2. Answer: 1/9 Please can you show me how to get the answer. I'm confused about joining the events of choosing 2 ...
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Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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24 views

Die Probability Question + Basics of Conditional Probability

A die is rolled twice. What is the probability of observing: a) a four and a three P (obtaining a four and a three) or P(obtaining a three and a four) therefore P(obtaining a four)* P(obtaining a ...
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24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
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What should the contestant do? [duplicate]

Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then one of the other ...
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1answer
21 views

How to make this bet fair?

A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the ...
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1answer
37 views

Invariant mesure of a reflected random walk

Let $(X_n), n \geq 0$ be a Reflected Random Walk defined by: $X_0 = 0$ and: $ X_{n+1}=\max( 0 , X_n + \xi )$ $\xi $ is a random variable such that $P(\xi=a)=\theta$ and $P(\xi=-b)=1-\theta$ for a ...
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1answer
21 views

How to compute of each player winning this sequence of games?

Players A and B play a sequence of independent games. Player A throws a die first and wins on a "six." If A fails, then player B throws and wins on a "five" or "six." If B fails, then A throws and ...
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3answers
38 views

How to compute these probabilities?

A pair of dice is cast until either the sum of seven or eight appears. How to compute the probability of a seven before an eight? Now, if this pair of dice is cast until a seven appears twice or ...
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47 views

How to compute this probability?

A drawer contains eight different pairs of socks. If six socks are drawn at random and without replacement, how to compute the probability that there is at least one matching pair among these six ...
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25 views

$ P[A] \leq P[A |\bar B] + P[B] $

How to prove that for any two events $A$ and $B$ $$ P[A] \leq P[A |\bar B] + P[B] $$ Can someone provide me hint how to proceed.
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How to establish the independence or otherwise of these compound events?

Suppose that $C_1$, $C_2$, $\ldots$, $C_n$ are mutually independent events in a sample space $S$. Then how to establish the independence or otherwise of these combinations of events? $C_1^c$ and ...
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35 views

Independence of two digits in a uniform$(0,1)$ random variable

I having troubles proving this: Let X be a Uniform$([0,1])$ distributed random variable. And let $X_n$ be the nth digit in the decimal expansion of $X$. Prove that if $n \neq m$ then $X_n$ and ...
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62 views

From distribution to Measure [duplicate]

I have been asked to create a new post with my question. So it is about starting from a distribution function and proving that we can always find a probability space. My attempt is this : So assume ...
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53 views

How to talk about a random variable which only exists on an event

Suppose I have some probability space, and some event $A$ which has probability close to, but not equal to $1$. Suppose that on this event, we can construct a random variable $X$. (For example, ...
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29 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
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1answer
32 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
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3answers
47 views

Central Limit Theorem vs. Weak Law of Large Numbers

So, just to begin with I feel like this is a problem I am massively overthinking, and the solution is very simple. That said, it has been a while since I've taken a math class, and so some of my ...
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1answer
34 views

Law of total expectation: $\mathbb{E}( X | A )$ and $p_{A|B}(a|b)$

I've got a question regarding the law of total expectation and conditional probabilities. Assuming I know the EV: $\mathbb{E}(X | A)$ and the PDF $p_{A|B}(a|b)$ (A,B are not independet but binomially ...
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1answer
40 views

Convergence in distribution/Distribution of X

For each $n = 1, 2, ....$, suppose that $X_n$ is a discrete random variable with range $\{1/n, 2/n, ..., 1\}$ and $\hspace{15mm}\mathrm{Pr}(X_n = j/n) = \frac{2j}{n(n+1)}$, $j = 1,...,n$. Does ...
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49 views

Prove the inequality for PDF which non decreasing in interval

I am working on the following problem. Given a pdf $f$ that is non-decreasing in the interval $ a \leq x \leq b $, show that for any $s>0$ $\int^b_{a}{x^{2s}f(x)}dx \geq ...
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45 views

Central Limit Theorem exercise question

Let $ (X_n)_{n \in \mathbb{N}}$ be i.d.d. random variables with $E{X_1}=0$, $Var(X_1)=1$ and $ S_n = X_1 + X_2 +...+ X_n $. Calculate $ \lim_{n \to +\infty}\Pr(S_n>\sqrt{n})$. On the back ...
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Transforming discrete R.V. to uniform R.V.

Suppose that I can generate some random variable $X$ that is distributed according to the CDF $F$. If $F$ is continuous then $F(X)$ is uniform $[0,1]$ (can anyone explain this to me). My question is ...
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42 views

Conditional Expectation: How to get $ \mathbb{E}( X | B ) $ from $ \mathbb{E}( X | A ) $ with $C=A+B$

I'd like to calculate some conditional expected values and I'm facing some problems. Here is what I've got: Known is: $ \mathbb{E}( X | A=a ) $ And I'd like to calculate: $\mathbb{E}( X | B=b )$ ...
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30 views

Reflected random walk

Suppose that $X_n$ is a reflected (in 0) random walk with parameter $\theta$. So $X_{n+1}-X_n = 1$ with probability $\theta$ , and -1 with probability $1-\theta$ when $X_n \geq 1$, if $X_n=0$ then ...
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48 views

Probability of intersection of non increasing sequence of events

Suppose that $(C_n)_{n\geqslant1}$ is a sequence of events such that $C_{n+1} \subset C_n$ for every $n\geqslant1$. Then how to prove the following probability statement? $$ \lim_{n \to \infty} ...
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4answers
49 views

What is the probability that at least one letter is in the correct envelope?

A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct ...
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28 views

How many bulbs should be inspected for probability to exceed $1/2$?

In a lot of $50$ lightbulbs, there are $2$ bad bulbs. How many bulbs should be examined so that the probability of finding at least $1$ bad bulb is at least $1/2$? My effort: Suppose $n$, where $0 ...
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40 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
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1answer
35 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
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Problem regarding Conditional probability

Let $\mathbf{X}$ be an $n-$ dimensional random variable. This variable can be written as $\mathbf{X} = \left[\mathbf{X}_1^T\hspace{5pt}\mathbf{X}_2^T\right]^T$. where, $\mathbf{X}_1$ is $m-$ ...
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Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
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Product of a discrete and absolutely continuous random variable, part deux

I have the independent random variables $U\sim N(0,1)$, $V\sim N(0,1)$, $W\sim b(1,1/2)$ I define $X=WU + (1-W)(V+1)$. I need to determine that $X$ is absolutely continuous, and determine a density ...
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20 views

Combining classifiers

How can I compute $P$($X$ | $f_0$, $f_1$, $f_2$), given the following pdf's: P($X$|$f_0$), P($X$|$f_1$).P($X$|$f_2$). I am interested in the case of discrete probabilities, but I guess it will be ...
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1answer
16 views

Product of a discrete and absolutely continuous random variable

I have the independent random variables $U \sim e(1)$ and $V \sim po(2)$. We define the variable $Y=UV$. I want to prove that Y is not absolutely continuous. The only hint I have recieved is that I ...
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Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
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1answer
58 views

Let F be a distribution function. Prove that X is a RV.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \ to \mathbb{R}$ by $X(\omega) = sup(y ...
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60 views

If a probability space has no measurable subsets with $P$ strictly between $0$ and $1$, then every random variable is constant a.s.

Let $(\Omega, \mathfrak{F}, P)$ be a probability space such that $\forall F \in \mathfrak{F}, P(F) = 0 \ or \ 1$. Show that for all random variables X on $(\Omega, \mathfrak{F}, P)$, $\exists \ c ...
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Most important results from pure math in applied probability?

I'm taking a course next semester at my university on applied probability (with relevance to signal and information theory). Although the nature of probability is mostly problem solving and applying ...
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56 views

If X and Y are random variables with the same distribution, prove that f(X) and f(Y) are random variables that have the same distribution.

Suppose X is a RV on $(\Omega, \mathfrak{F}, P)$. Let f be Borel-measurable on $(\mathbb{R}, \mathfrak{B})$. 1 Show that f(X) is also a RV on $(\Omega, \mathfrak{F}, P)$. 2 Let Y ba RV on ...