Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
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Stopping time for uniform law
Let $X_1, X_2, \dots$ be IID Unif$(0,1)$ random variables and let $N=\min \{n : S_n=X_1 + \dots + X_n > \ln(2) \}$. Find the expectation of $N$.
I've tried three approaches.
First I showed that $...
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Joint density of a bounded random vector
Let $X = (X_1, \dots, X_n)$ be a random vector with support $\mathbb{R}^n$, and with distribution $F_X(x_1, \dots, x_n)$ and density $f_X(x_1,\dots, x_n)$. Consider the bounded transformation of $X$ ...
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Exercise 2.8 from Billingsley
In Billingsley's Convergence of Probability Measures, Exercise 2.8 asks:
Suppose $\delta_{x_n}\Rightarrow P$. Then $P=\delta_x$ for some $x$.
Here, $\delta_{x}(A)=\textbf{1}_A(x)$ is the unit mass. ...
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binomial distribution but sometimes the last outcome doesn't matter
Here's the motivation for my question: I'm designing an RPG. To simplify as much as possible, lets say my enemy has $h = 4$ HP and I deal $a = 1$ damage with every attack.
However, there's also a $p$ ...
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Express a set as a union of two disjoint sets [closed]
Suppose that $A$ and $B$ are subsets of $S$.
a) Express $A$ as a union of two disjoint sets.
b) Express $A \cup B$ as a union of three mutually exclusive sets.
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Lottery ticket contains 6 numbers, find exactly 4 of the 6 winning numbers, with no repeat.
Lottery ticked has 6 numbers each from 1-49, with no repeats. Find the probability of matching exactly 4 of the 6 numbers if the winning numbers are all randomly chosen.
My attempt:
There are ${6 \...
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Prove that discrete first hitting time is a stopping time
I have problems with the proof that a first hitting time is a stopping time:
Let $\tau$ be the first hitting time into the set A, for a process $\{ X_n \}$ adapted to a filtration $\mathcal F_n$.
I ...
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Polya urn scheme by induction task
I've been trying to solve the task given in "An Introduction to Probability Theory and Its Applications" by William Feller. The task is to show by induction that the probability of a black ...
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Given $P(B\mid A)=0.4$, $P(A \cap B)=0.1$, and $P(B^c)=3P(A)$, what is $P(A \cup B)$? [closed]
Can you help me to solve this question please?
Given $$P(B\mid A) = 0.4\quad \quad P(A \cap B) = 0.1 \quad \quad P(B^c) = 3P(A)$$ what is $P(A \cup B)$?
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on the Thompson sampling proof
I was reading recently the Thompson sampling paper https://arxiv.org/pdf/1205.4217. The non-constant (w.r.t $T$) leading term of the regret is obtained (begining of page 5) by bounding $\sum_{t=1}^T\...
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Moment method and central limit theorem
Consider functions $P_n$ depending on a parameter $n$, and a fixed function $\phi$. Consider also a discrete set $D$. Assume we have the convergence
$$
\sum_{d \in D} P_n(d)^k \phi(d) \underset{n \to \...
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Creating an Estimator for the Dimension of Bernoulli-distributed Vectors from Observed pairwise Dot Products
I have I individuals defined by vectors $P_i \sim \mathcal{B}(1,1/2)^d$ iid. We can note $\overline{P}_i = \langle P_i, \textbf{1} \rangle$ the proportion of 1's in individual i; $c_{ij} = \langle P_i,...
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Probability Question with Deck of Cards - three players, 5 cards each, P(at least one person has exactly two aces)
Three players are each dealt, in a random manner, five cards from a deck containing 52 cards. Four of the 52 cards are aces. Find the probability that at least one person receives exactly two aces in ...
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exponentially decaying weighted integral lower bound
For $f\in L^2$ for example, can we achieve sharpest lower/upper bound for the following integral for small $\epsilon>0$
$$\int_{0}^{T}e^{-\dfrac{t}{\epsilon}} |f(t)|dt $$ and
under what conditions ...
