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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Arguments about convergence of sequence of random variables

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
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1answer
19 views

Probability of a number in the real line

I have read that the probability to pick a rational number in the real line is null. My problem is: If $S$ is a dense set in the real line, what is the probability to pick an element of $S$? There ...
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1answer
10 views

The $k$-th order statistic of $U(0,1)$ are Beta?

In Wikipedia it says that: Let $X_1,\dots, X_n$ be an i.i.d. sample from $U(0,1)$. Let $X_{(k)}$ be the $k$-th order statistic from this sample. Then the probability distribution of $X_{(k)}$ is ...
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0answers
7 views

Comparing two hitting times of Bessel process

Suppose $X$ is a Bessel process of dimension $1 < d \le 3$ with $X_0 = 0$. Then $X$ satisfies the SDE $ dX_t = \frac{d - 1}{2X_t} dt + d W_t$ for some Brownian motion $W_t$. Let $a > 0$. Let ...
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Use Chebyshev’s inequality to choose $n$ such that $P(\bar{X} > 4) > 0.9$

Use Chebyshev’s inequality to choose n such that $$ P(\bar{X_n} > 4) > 0.9 $$ where $$ E[\bar{X_n}] = 5 \ \ \ \ \ Var[\bar{X_n}] = \frac{4}{n} $$ The problem I am having when using Chebyshev's ...
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27 views

i have done partial of this problem, need help on this

Problem: Start with initial capital $ x$ and consider a continuous time game of betting $1$ on the outcome of the Brownian motion Wt started at x at time 0. Namely, Wt=x+Bt = fortune at time t, where ...
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1answer
63 views

Why do we like sticking random variables into their own distributions?

Let $X$ be a random variable taking values in the set $S$. It has some distribution $f(s)$. Often in statistics, we are interested in the real valued random variable $f(X)$. Here are some examples: ...
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1answer
51 views

Does it pay to know what you know?

Let's play a game. I ask you question a yes/no question, and you answer. You don't answer with a yes or no though, you answer with a probability of it being yes ($P \in (0,1)$). For example, I might ...
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1answer
146 views

Probability of two adjacent seats at a round table being available

There are Fifteen seats at a round table. There are three people already seated, their locations chosen uniformly at random. Three people wish to join the table and sit next to each other. What is ...
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1answer
20 views

Probability number chosen at random from S will be odd

The question is: Let $S = \{1,2,3,\ldots\}$. Let $P$ be a probability measure defined by $P(\{n\}) = 2(\frac{1}{3})^n$ for all $n$ in $S$. What is the probability that a number chosen at random from ...
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1answer
16 views

Does the Information Gain algorithm favor a high-entropy attribute or a low-entropy one?

This might not be mutual to mathematics but it does relate to Information-Theory. My question is: Does the InformationGain algorithm, in Decision-Tree machine-learning, favor a high-entropy ...
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2answers
37 views

Doubt in Conditional Probability

I'm studying Information theory from the book Information Theory, Coding and Cryptography-Rajan Bose. I got confused at one pos where they have derived the equation ...
2
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1answer
172 views

Number of arrangements of $n$ couples around a circular table with restriction

A group of $n$ couples (a total of $2n$ people) sit at a circular table. Arrangements that differ by any rotation of the seating positions are considered to be the same. Find a formula for the number ...
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1answer
17 views

Should monotone convergence theorem say uniformly bounded?

saz pointed out to me the difference between bounded and uniformly bounded: $Y$ is uniformly bounded: there exists $C>0$ such that $|Y_n| \leq C$ for all $n \in \mathbb{N}$, i.e. $$|Y_n| ...
2
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1answer
65 views

Number of Isolated Edges in G(n,p)

I am attempting to find the number of isolated edges in the Erdos - Renyi graph G(n,p). I need to find the formula for the expected number of isolated edges. I've broken the equation down into ...
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1answer
15 views

$0 \leq Y \leq M$ random variable, $p > 1$. Calculate $\mathbb{E}(Y^p)$

$0 \leq Y \leq M$ random variable, $p > 1$. Show that $\mathbb{E}[Y^p] = \int_0^M py^{p-1}\mathbb{P}[Y \geq y] dy$ My attempt: $\mathbb{E}[Y^p] = \int_0^{\infty} Y d\mathbb{P} = \int_0^{M} Y ...
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1answer
438 views

Where can I find the solutions to exercises of Probabilistic Graphical Models?

I am self-learning Probabilistic Graphical Models written by Daphne Koller. And for testing how well I learned, I did the exercises in the textbook. But I have no solutions to these exercises. Can ...
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0answers
10 views

On a Probability notation

What could mean this notation : $\mathbb{E}[X(.)|\mathcal{F}]_G$ ? where G : $\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega,P, \mathcal{F})$. X could be a ...
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1answer
23 views

Simulated two random variable $Z_{1}$ and $Z_{2}$ such that $Z_{1}\sim N(0,\sigma^{2})$ and $Z_{2}\sim N\left(0,\frac{(\sigma^{2})^{3}}{3}\right)$

How I can, by the central limit theorem, simulated two random variable $Z_{1}$ and $Z_{2}$ such that $$Z_{1}\sim N(0,\sigma^{2})$$ $$Z_{2}\sim N\left(0,\dfrac{(\sigma^{2})^{3}}{3}\right)$$ And ...
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1answer
24 views

Even moments of distribution given probability density function

Given the probability density function $f(x)$, and the $𝔼[X] = \frac{2}{\sqrt{\pi\lambda}} $, how best should I go about deducing the even moments of this distribution? $f(x) = ...
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1answer
319 views

Belief Propagation Algorithms for Graphical Models with Cycles?

Belief propagation algorithms cannot solve for the probabilities of a cyclic graphical model; they only work for acyclic graphical models. For undirected graphical models (for example Markov random ...
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0answers
28 views

Alpha representation of probability distribution

Probability vector is an $n$-dimensional vector $p=(p_1,...,\ p_n)$ that the sum of whose components equals one, i.e. $p_1+...+p_n=1$. If we take the square root of each component of probability, we ...
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159 views

Conditional probability given two variables from conditional probabilities of one variable

I have a question which comes from an example of a textbook, but all I am concerned with is how we go from having probabilities $P(X|A)$ and $P(X|B)$ to having $P(X|A,B)$. In the example events $A$ ...
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0answers
13 views

Expected Value of $P(|Y_n^{(K)}| > \epsilon)$ where $Y_n^{(K)}$ is the random sum of a sequence of RV converging to 0 in Probability

I have been struggling with this for countless hours, I would appreciate a hint to get me going in the right direction (no complete answer please) Problem: Assume that for all $k \in \mathbb{N}$ ...
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1answer
61 views

Derive probability relation between Bayesian Network leaves

Suppose that we have a network like the one in figure in which we know all the conditional probabilities $P(A_i | B_j)$. Is it possible to compute the two conditional probability $P(B1 | B2)$ and ...
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1answer
70 views

Sources to learn and understand advanced probability in ML models

Could someone please suggest some good positions on probability (and perhaps statistics)? My ultimate goal would be to learn and understand Machine Learning and its models, such as Neural Network, ...
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0answers
111 views

Soundness and Completeness of the d-separation

I was reading Koller's book on Probabilistic Graphical Models and was wondering how to prove soundness and completeness of the d-separation ? Let me explain what is d-separation and it's properties ...
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2answers
2k views

Finding Probability of P(S|W) at Bayesian Network of Rain Problem

I am studying Bayesian Networks. Given that variables: $W$: Wet grass $R$: Rain $S$: Sprinkler I know the probabilities of: $P(C)$ $P(S | C)$ $P(S | !C)$ $P(R | C)$ $P(R | !C)$ $P(W | R,S)$ $P(W | ...
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1answer
1k views

Extended Bayes' theorem: p(A | B, C, D) - Constructing a Bayesian Network

I'm having some difficulty understanding Bayes' theorem with multiple events. I'm trying to put together a Bayesian network. I have four independent probabilities but I have found that A, B and C can ...
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1answer
613 views

sum out a random variable

I have a bayesian network like that: B -> A <- F And this are the values for A: P(A=true | B=true, F=false) = 0.01 P(A=true | B=true, F=true) = 0.92 P(A=true | B=false, F=false) = 1.00 ...
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2answers
394 views

Joint probabilities and conditional independence

I'm going through a revision paper and looking at the solutions and I come across this. Given a Bayesian Network (sorry I cannot post images): $A$ and $B$ are parents of $C$. And $C$ is parents of ...
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138 views

How do I prove and expand Bayesian Networks?

Attempting to understand Exercise 20 (pdf page 44) in the paper: (Warning: large paper; small exercise) Bayesian Reasoning and Machine Learning The party animal problem corresponds to the ...
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1answer
47 views

A question about Bayesian Networks from Judea Pearl's book.

"Given a probability distribution $P(x_1, \dots, x_n)$ and any ordering d of the variables, the DAG(directed acyclic graph) created by designating as parents of $X_i$ any minimal set П$_{X_i}$ of ...
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1answer
66 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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2answers
191 views

In Bayesian network graphs, how to systematically search for conditional independent nodes?

In Bayesian network graphs, how to systematically search for pairs of conditional independent nodes, and the associated condition node(s)? Is there some simple rules or algorithms to follow?
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81 views

Conditional independence in Bayesian network with qualitative influences

I have some troubles solving an exercise from the book Probabilistic Graphical Models (pgm.stanford.edu). We are given the bayesian network with binary-valued variables. We do not know the CPDs, ...
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1answer
45 views

conditional probability question Given bayesian network could not understand solution given

Given Bayesian network can't understand the two last steps in why the p(C=c|E=e,~H) can get out of the e sum? and why sum p(E=e|A,S,~H) and sum p(C=c|E=e,~H) and been neglected? Thank for the ...
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1answer
143 views

Checking independence of variables in a Bayesian network

I need a little help with Bayesian Networks. Consider given the following network (all variables are binary) and we need to check conditional independence of $A$ and $C$ if $X$ and $Z$ are given. Any ...
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1answer
52 views

Bayesian network query

I am having a bit of trouble with something that I imagine is fairly easy. I am wondering how to get the probability of alarm, JohnCalls, and MaryCalls if they have no prior knowledge of their ...
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1answer
24 views

Confusion in a simple Bayesian Network

I am studying about Bayesian Networks for the first time from a particular source. It gives the example of the following Bayesian Network Now I understand this network, then they go on with ...
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1answer
63 views

probability calculation for bayesian network

I am studying Bayesian belief networks and in that I am struggling to understand how probabilities are calculated. I found this article here and the network is this: The associated probabilities ...
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0answers
155 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
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Bayesian networks: calculating likelihood of data dgiven probabilities and network structure

I'm trying to understand how to calculate the likelihood of a combination of binary variables given the network structure and probabilities seen here: Given a bayesian network as follows: ...
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1answer
271 views

Bayesian Network, Sprinkler Example

In reference to the wet grass / sprinkler Bayesian network problem at this site: http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html Pr(S=1 | W=1) has been determined as 0.430. Could someone please ...
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1answer
175 views

Bayesian Network - unclear homework example

I am not sure if it is me or the example: A doctor gives a patient a drug dependent on their age and gender. The patient has a probability to recover depending on whether s/he receives the drug, ...
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2answers
101 views

Conditional Independence - Bayesian Network

May the probability distribution $ P(A,B,C,D) $ given as: $ P(A,B,C,D) = P(A)P(B)P(C|A,B)P(D|C) $ The task is to show that this holds $ A \bot B | \emptyset $ and $A\bot D|C$. First thing I'd like ...
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2answers
27 views

Bayesian Network Probability involving intersection

Imagine a node "I" with two children, "W" and "H". "I" means that roads are icy, and "W" means that Watson crashes. "H" means Holmes crashes. If I wanted to know the probability of the roads being ...
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1answer
30 views

Probability issue given a Bayesian Network [on hold]

If we have a Bayesian Network A -> B ->C then P(B|A, C) = P(B|A)? Thanks!
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1answer
50 views

Bayesian network and unknown probability

I'm trying to solve questions regarding bayesian network, and now I was wondering if it is possible to know the probability of an unknown variable in the tree. For instance, I have this tree, ...
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1answer
193 views

Probabilities from bayesian network

I am doing problems related to bayesian network. After reading the theory part I am able to understand that by making a network or reducing a problem to some bayesian network, we are simplifying a ...