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Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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80 pick 20, sort asd, sum per 4 numbers, why result a bit bias towards even number

I am working on the probabilities of a game where 1:they pick 20 unique number(non-repeating) from range 1 to 80 2:sort those 20 number in ascending order 3:sum every 4 number, take the last digit ...
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80 views

Calculate variance the right way with two random variables

I'm currently assigning a introductory stats class, and I just can't seem to find out when to use the different variance identities. I have provided an example of an assignment where I got it wrong, ...
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Convergence of regression coefficients to probability density

By simulation we create a vector $Y = (y_1,y_2,...,y_n)$, where each $y_i \in R$ is independently drawn from a given non-degenerate distribution. Next we create by simulation a vector $\xi = (\xi_1,\...
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Computing the probability of winning the grand prize in a hypothetical game

I'm a beginner to probability so please be a bit gentle on me. I have trouble finding the correct approach / search terms to find related problems. I face the following problem. In a hypothetical ...
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Joint distribution of the magnitude/phase of a circular bivariate normal distribution?

A bivariate normal distribution with no correlation and identical variance in both dimensions can be written as $$ P(x,y|\mu_x, \mu_y, \sigma) = \frac{1}{2\pi\sigma^2}\exp{\left(-\frac{1}{2}\left(\...
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How do Bayesians verify their methods using Monte Carlo simulation methods?

Background: I have a PhD in social psychology, where theoretical statistics and math were barely covered in my quantitative coursework. Through undergrad and grad school, I was taught (much like many ...
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1answer
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Variance of X+Y upper bound

I have that $Var(X)=0.5$ and $Var(Y)=0.02$ and I am being asked for the maximum value $Var(X+5Y)$. I cannot assume that $X$ and $Y$ are independent variables. It can be shown that $Var(X+5Y) = 1 + ...
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Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?

It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
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Bernoulli essays r [on hold]

Consider a fair coin. Let $H_n=0$ if you observe tail and $H_n=1$ is you observe face. Then $P(H_n=0)=P(H_n=1)=\frac 12$. The distribution of $S_n=H_1+\cdots+H_n$ is binomial with parameters $n$ and $...
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If $X\sim \operatorname{lognormal}$ then $Y:=(X-d\mid x\geq d)$ has approximately a Generalized Pareto distribution

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d\mid x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: ...
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Find marginal distribution

The random vector $(X,Y)$ is uniformly distributed over $$D=\{(x,y): 0 \leq x \leq 2 , 0 \leq y \leq 2-x\}.$$ Find the marginal distribution of the random variables $X$ and $Y$. For the radom vector $...
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Finding probability bucket which result in 90% correct classification

I have a dataframe of two columns, one of which contains probabilities of event X happening and the second column is whether or not X did occur as indicated by a 0,1. I would like to find the buckets ...
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How does the probability of events change if an event does not occur

Suppose that someone tells me I will collect $\$100$ dollars within some time interval. Those time intervals are 1 to 7 days, 8 to 30 days and eventually after 30 days. Let $A$ be the event I ...
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Stochastic Sequence; Compute $\lim_{n\to\infty}$

Let $(X_n)n≥1$ be a stochastic sequence of i.i.d. random variables, each $X_n$ with values in the set {1, 4, 8, 16} and probability distribution: $P[X_n = 1] = 1/6, P[X_n = 4] = 1/4, P[X_n = 8] = 1/3,...
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Determining a probability distribution from constraints on where its mass is

Let $X$ be a random variable over the real line. Suppose that we know that $X$ is a Pearson distribution. Furthermore, suppose we know how the mass of $X$ is distributed into 6 intervals, so that if $...
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1answer
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Number of tests to perform to get confidence rate

If I have an issue that causes a failure at a rate of, say, 1 in 20 cycles and then deploy a fix, how do I work out how many test cycles to perform to be confident to 95% that the fix worked? I know ...
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Tail generating functions and PGFs [on hold]

Hi I'm not sure where to start for the following question. I don't know how to find fx(X). Could someone please help me get started, the lecture notes I've been given don't really help me. Thanks
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Deriving Value Function of a Markov Reward Process

I am looking through the coursework for a Reinforcement Learning course (I am not enrolled in it, this is for my own study). In the lecture notes, on page 6, equation 11, they provide the following ...
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Probability of getting two $z$-scores that are within one standard deviation of each other

Randomly choose two $z$-scores following a normal distribution with mean 0 and standard deviation 1.* What is the probability that the $z$-scores are within one standard deviation of each other?
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Likelihood of an event happening

Let's say that I had a test with three questions, and the probability of answering the questions right was .5, .4 and .33 respectively, presuming that the probabilities of each question are ...
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Modelling values by overlaying existing distributions

Imagine I have two datasets, one has values of a dependent variable, time spend walking, along with many other independent variables such as for instance gender, age group, day of week and dog owner ...
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Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$

How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
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1answer
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Multi-armed Bandit :a lower bound for the expected sample size from an inferior population

I am reading Asymptotically efficient adaptive allocation rules to study the multi-armed bandit problem, and have a question. The question is about the proof of Theorem 2. THEOREM 2. Assume that $...
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“marginal standardisation” for comparing predicted probabilities between two groups? [on hold]

I have a doubt regarding to the R package "margins". I'm estimating a logistic model: ...
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9 views

dirichlet distribution and excessively large numerator

what I am trying to do is calculating posterior probability using dirichlet distribution as my prior. the situation is like this. a web log have three variables A, B, C, and each variable's value is ...
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Why is the marginal distribution/marginal probability described as “marginal”?

Marginal generally refers to something that's a small effect, something that's on the outside of a bigger system. It tends to diminish the importance of whatever is described as "marginal". So how ...
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No population? How to answer such question?

Say the givens are: probability of a battery being defective is = 0.40 being not defective = 0.60 STD = 1.5 hours mean = 7.5 hours The question states: if 5 batteries are randomly selected what is ...
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1answer
38 views

Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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What is the relation between message passing and probabilities in Bayesian inference?

The belief propagation algorithm is a message passing algorithm that can be used to estimate marginal probabilities on Bayesian networks. What is the definition of these messages? What is the ...
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I have a scatter plot of a probability Function against a variable called Luminosity and I want to calculate its Mean and Standard deviation [closed]

Hello, beginners question: I have two lists of data in a Python program. 1) [Luminosity Model} - The luminosity or brightness of stars in the galaxy, and 2) [Prob_LTlogz] - Probability distribution ...
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Assigning probabilities to ensemble experts (classification)

Suppose we have a set of experts which predict on a data set, and the true labels are also given. I would like to find out the probabilities for combining predictions of separate experts. So the ...
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1answer
34 views

pdf from a set of conditional pdfs

I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
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2answers
43 views

Identify a switch between probability distributions

Say I have two normal distributions with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$, respectively. A t-test reveals that the means are significantly different, but the ...
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1answer
39 views

How to calculate poisson approximation?

I have a p-value definition as follows and I would like to implement it. The values of n1, n2, n3, n4, n5 and N are as follows. n1 = 102 n2 = 95 Calculating λ: ...
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Particle filter for diagnosis

I have two annual measurements taken on medical images depicting a lung cancer tumor 's condition. I have likelihood function that taken in the measurement values and estimates malignancy of the tumor....
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expected value (Sheldon Ross 9th edition)

A person buy 10 lottery tickets with each having a winning probability p. If he wins a prize in atleast one of the tickets,he gets addicted and will keep on buying tickets till he wins again. Find the ...
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30 views

variance of multinomial distribution

Assume $A_{kj} \sim$Multinomial$(1, \;\underbrace{(1/m, 1/m, ..., 1/m)}_{\textrm{m times}})$, where $k=1,2, ... m$ and $j=1,2, ... n$. It is clear to see that $\sum_{k=1}^mA_{kj}=1$. If we impose a ...
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variance of restricted random vars

Let $X_1$ and $X_2$ are i.i.d. with Bernoulli($p$). If we are interested only on a set of realizations of $X_1$ and $X_2$ denoted by $Y_1$ and $Y_2$, respectively, such that $Y_1+Y_2=S$, then what ...
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39 views

Density Estimation Efficieny

My Question Let's say a set training samples like D from a discrete distribution like p(x) over a discrete variable vector like x is available. We don't have any prior knowledge about the form of p(x)...
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14 views

Draw likelihood graph from a probability distribution [closed]

I'm writing an algorithm that adds a special node of a tree with probability: p = (1 - 0.96^d) where d is the depth of the tree. I want to draw a graph using python that shows for which values of d ...
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56 views

maximum of two independent mass functions (PMF) [closed]

If Y = max(N1' , N2'). how to calculate the PMF of Y? Discretized normals can we use this formula? Or similar procedure? $F_1(y) * f_2(y) + F_2(y) *f_1(x) $ Any help will be greatly appreciated. ...
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Combining probability of default of two variable

Lets just say I have probability of defaults (PD) from two different Macroeconomic variable (MEV) with 3 different scenarios like in the picture below. What I am trying to determine is the combined ...
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1answer
38 views

How to convert classification features into a ranking function?

I am using 3 features (x1, x2, x3) for binary classification. All my feature values are in 0 to 1 range (unit range). I obtained how important each feature was in classification as follows (i.e. <...
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Is a transformation of a conditional distribution identical to the conditional of the transformation?

Problem Say that we can find a random vector $(Y_1, \cdots, Y_n)$ whose distribution is identical to the conditional distribution of $(X_1, \cdots, X_n)$ under $f(X_1, \cdots, X_n) \le 1$, where $f(\...
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Can someone explain the concept of parameter didactically to me? [closed]

Let $\mathcal{P}$ denote a family of probability distributions on $\textbf{R}^{n}$. We then use a parametrization to describe $\mathcal{P}$, that is, a map $\theta\rightarrow\mathcal{P}_{\theta}$ from ...
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34 views

Showing $Y_n\stackrel{p}\to Z$ where $Y_n=B_nZ+(1-B_n)X$

I am reviewing some of my old class notes again, and I came across the following problem. I think I have solved the problem correctly, but I wanted to see what others here thought. Do you think I ...
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63 views

Conditional probabilities for sequence of events

The following table represents all possible paths of dichotomous events at 5 time moments. At each time moment either 1 or -1 event occurs with probabilities $p$ and $q$. Time stops when one observes ...
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1answer
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Same Example for Two Counter Examples

I'm learning some probability theory and I've come across the following: For an example of a sequence of random vairables that converges in the mean square sense but not almost surely: We set $$P(X_n=...
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19 views

probability density of a random vector greater than some value? [duplicate]

In single dimension, the probability that a random variable $X$ is greater than some value $x$ is easily related to the cumulative distribution(c.d.f.) as $Pr(X > x) = 1 - F(x)$ if only $Pr[X \leq ...
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calculate probability of people coming to an event based on historic data

I have a system of recording attendees to a conference that happens every month. These attendees go to the conference each month. However, some come to the conference, and some don't. Based on 12 ...