Questions tagged [probability]
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For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.
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conditional probability question doubt
Consider an experiment having three possible outcomes that occur with probabilities $p_1$, $p_2$, and $p_3$, respectively. Suppose n independent trials of the experiment are conducted and let $X_i$ ... probability probability-theory random-variables conditional-probability- 3
Impossibility of probability measure on $2^{[0,1]}$
Let $\Omega = [0, 1]$ and define a set function on the subsets $(a, b] \subset \Omega$ as $P( (a, b] ) = b-a$ Prove that no extension of $P$ from the subsets where it is defined to the power set of $\... probability probability-theory measure-theory lebesgue-measure- 49
Calculating the probability of one approach outperforming another approach
I'm trying to calculate a probability for some outcome, but am running into some issues. I did take a math class that involved probability calculations in high school but it's been some time and I can'... probability problem-solving- 101
Proof verification for independence using characteristic functions
Let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be independent random vectors. We need to show that for $s=(s_1,s_2,s_3,s_4)^T$, $s_1X_1+s_2X_2$ and $s_3Y_1+s_4Y_2$ are independent. Proof: Let $W=(X_1,X_2,Y_1,... probability probability-theory independence- 786
how does Bertrand's paradox challenge the classical definition of probability?
On page 9 of Papoulis's book[Probability, Random Variables, and Stochastic Processes], the classical definition of probability is as follows: The probability of an event equals the ratio of its ... probability probability-theory paradoxes- 23
Conditional expectation on squared sum of independent random variables
Given $X$ and $Y$ independent random variables of means $0$ and variance equal to $\sigma^2$, and $Z = X + Y$, find the conditional expectation $E[Z^2|X = x]$ for any value $x$ where the conditional ... probability statistics statistical-inference conditional-expectation- 82
Does almost surely convergence implies convergence in $L^p$?
I have the following excercise: Does almost surely convergence imply convergence in $L^p$? Prove or give a counterexample. For $0 < p < ∞$, we say $X_n$ converges in $L^p$ to $X$ if $X ∈ L^p(Ω)$ ... probability probability-theory convergence-divergence- 13
Probability of markov chain in a finite set
Let's $X$ be an homogeneous Markov chain with three states $\{1,2,3\}$ et denote $(\pi_1,\,\pi_2,\,\pi_3)$ the initial probabilities and $P=(p_{ij})_{1\leq i,j\leq3}$ the transition matrix. Let's ... probability combinatorics markov-chains- 440
Find the marginal density of $Y$ and $\mathbb{P}(Y>X)=\frac{5}{8}$.
The random vector $(X,Y)$ has joint density function given by $f(x,y)=\frac{1}{4}(x+3y)e^{-(x+y)}$ if $x,y\geqslant0$ and $f(x,y)=0$ otherwise. Find the marginal density of $Y$ and $\mathbb{P}(Y>X)=... probability probability-theory probability-distributions solution-verification- 837
What is the best interval size to consider in calculating the probabilities of a normally distributed sample?
I have a sample of size 80 which is normally distributed. I want to calculate the probabilities on different intervals. Is there any way or method of finding the most optimum interval size? Meaning ... probability normal-distribution- 3
Interesting probability problem on infinite processes
We start with a binary string of length $2a>0$ which contains exactly $a$ zeros and $a$ ones and we call it $L_0$. For every non-negative whole number $n$, we form $L_{n+1}$ by picking a random bit ... probability- 577
Parameters in survival function not changing output.
I am looking at this paper, and they are modeling "cohesiveness" in a group by doing the following: First, they define a "common action" of all individuals in a group by using the ... probability random-variables parametrization- 256
If set $C$ has 60 elements, and set $B_i$ has 9 elements randomly sampled from $C$, what is the mean value of $i$ to seeing all elements of $C$?
This question is inspired by the Yu-Gi-Oh TCG. Every so often, the game receives a new set containing either 60 or 100 new cards (this is the set $C$ in the question title). However, they don't just ... probability statistics- 343
Probability of rolling more 1's or 2's than 6's on X dice
I'm trying to figure out what the odds are for rolling more 1's and 2's than 6's (on standard dice), according to the number of dice rolled. For one die it's obviously 33.3% What about for 2 dice ... probability dice- 123
Strong mixing of a function of strong mixing and convergence sequence
Suppose the process $X = \left\{X_{t}:t\in Z\right\}$ is strong mixing with the coefficient $\alpha(j) \rightarrow 0$ defined as \begin{equation} \alpha(j)=\sup_T\sup_{1\leq k\leq T-j}\sup\{\lvert P(A\... probability convergence-divergence mixing- 440
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