Questions tagged [probability]

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

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23 views

how to multiply two conditional probabilities in general

I am trying to understand how to multiply two conditional probabilities. $P(X|C) \times P(C| P,S)$ seems to equal to $P(X,C | P,S)$. How to understand this? I understand the product rule, but how ...
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21 views

What is the distribution of the sum of a Log-Normal variable and a Logistic variable?

I was wondering if there is any known distribution that is the sum of a Log-Normal variable and a Logistic variable is? What about the sum of Log-Logistic and Logistic? We can assume the variables ...
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35 views

Conditional Covariance Problem

Suppose we have independent (not necessarily identical) normally distributed random variables X, Y. If we're given that, upon sampling each variable, X is some multiple a of Y (i.e. x = ay), what is ...
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1answer
30 views

bivariate transformation when U=X/Y and Y=0

I am considering using transformation U=X/Y and V=Y (X,Y are iid distributions defined on real line). Almost all the textbooks I have read did not consider the case when Y=0, at which U is undefined. ...
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29 views

converting tanh activation output to a probability

I am trying to implement $PPO$ for continuous action spaces so need probability of taking actions from a neural network with a tanh activation in the output layer since the action space ranges from $[-...
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1answer
47 views

Inverse distribution of gaussian mixture

In one of the papers I've encountered, the authors propose a copula function $$ c(u_1, \ldots, u_d; \Theta) = \frac{\psi(y_1, \ldots, y_d; \Theta)}{\prod_{j=1}^{d}\psi_j (y_j)}$$ where $\psi(y_1, \...
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4answers
124 views

Probability of absolute value of a sum of random variables

Consider two random variables $X$ and $Y$, and let $b$ be a real number. Show that $$P(\mid X+Y\mid>b)\leq P(\mid X+Y\mid>b,\mid X\mid>b/2)+P(\mid X+Y\mid>b,\mid Y\mid>b/2).$$ I'm ...
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33 views

Find confidence interval of mean without knowing sample size?

In our calculation, we have only $\mu$ and $\sigma$ values. Is that possible to find the confidence interval of $\mu$ and $\sigma$, without knowing the sample size? As per our idea, we can calculated ...
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26 views

Probability of association in logistic regression models

I’m using a logit model to evaluate the association of a set of habitats (H) to a particular land cover. Both land cover and H are binary variables. ...
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27 views

How to derive a ranking function from observations

I have a employee dataset with the following 5 details. projects completed customer ratings number of bugs reported customer complaints profit I want to rank ...
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10 views

Predict Taking a Loan

I want to build a Logistic Regression model to help me predict taking a loan by a customer base on several variables. I started building a data set for the customers who already took loans and some ...
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1answer
26 views

Unusual Markov inequality for normal distribution

I'm trying to answer the following question from Larry Wassermans book on statistical inference. My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
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24 views

survival probability to proportional hazard ratios in r

I am using Cox proportional hazard regression to determine the association between physical behaviors (physical activity, sedentary behavior, and sleep) and mortality. to elaborate on these results, I ...
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24 views

Bayesian chi-squared tests

I have a dataset with two groups of participants. Each participant performed a repeated measures task on which three types of errors could be made. I want to measure the difference in distributions of ...
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1answer
87 views

KL divergence between gaussian and uniform distribution

Is the KL divergence not defined because uniform has bounded support and gaussian has unbounded support? How else can I calculate the distance of my gaussian to a 'maximum entropy' distribution if I ...
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2answers
95 views

Probability of seeing a bird on a certain date based on historical notes

I have a database filled with different bird species that were seen on different dates (10 years of records). Each row in the table contains: Date, Time, Bird Species, Spot where it was seen So it ...
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1answer
45 views

Why does discrete data distribution has differential entropy of negative infinity?

Recently I've been reading a paper. In section 3.1, it says "Since the discrete data distribution has differential entropy of negative infinity, this can lead to arbitrary high likelihoods even on ...
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1answer
74 views

Why is this probability distribution biased towards even numbers?

I am working on the probabilities of a game where they pick 20 unique number(non-repeating) from range 1 to 80 sort those 20 numbers in ascending order sum every 4 numbers and take the last digit as ...
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3answers
218 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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23 views

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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28 views

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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1answer
34 views

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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2answers
156 views

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
21 views

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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1answer
56 views

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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1answer
154 views

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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1answer
17 views

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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12 views

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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2answers
289 views

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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1answer
68 views

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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30 views

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
30 views

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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20 views

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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29 views

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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2answers
27 views

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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25 views

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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1answer
160 views

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
22 views

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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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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1k views

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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1answer
42 views

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
45 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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26 views

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
71 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
48 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
56 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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40 views

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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30 views

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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35 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 ...