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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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Is this Markov Chain calculation correct?

$S=\{1,2\}$ $\alpha = (1/2, 1/2)$ $P= \begin{bmatrix} 1/2&1/2\\ 0&1\end{bmatrix} $ Find $P(X_1=1 | X_0=1)$ Given solution: $P(X_1=1 | X_0=1) = \frac{P(X_1=1 , X_0=1)}{P(...
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45 views

Distribution of variable sampled from bounded distribution?

First time asking question but have learned a lot reading through previous posts :) Say I have a gym with 10000 members, so each day I will have a distribution of their time spent in the gym. This is ...
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36 views

Distribution to model Binomial distribution with parameter p in trial n dependent on result from trial n-1?

I'm wondering how one can model a Binomial distribution as described in question. E.g., p = 0.5 for trial n = 0; p(n+1) = p(n) + 0.01 if for trial n Bernoulli(p(n)) samples to 1, else p(n+1) = p(n) - ...
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27 views

Application of law of iterated expectations

I would like your help to show a statement that uses the law of iterated expectations. In my notation $Supp_X$ denotes the support of a random variable $X$. Consider the random variables $\epsilon,...
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31 views

Bayesian updating - update probability that measurement is real

I have a sequence of observations, which are either a measurement of 'active' or 'inactive'. These measurements are not necessarily accurate, with false negatives being more unlikely than false ...
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23 views

I.i.d.-ness of some functions of random variables

I have some doubts on the i.i.d.'ness of some functions of random variables. The framework What I'm describing below is a simplied version of a well known model in economics of demand and supply. ...
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41 views

Have a question about bivariate random vector [closed]

When X follows beta (2,2) and X = x, the conditional distribution of Y is the binomial distribution B (15, x). The probability mass function P (Y = y) of E (X), Var (X), E (Y), and Var (Y) obtained by ...
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37 views

Which is faster: a bank with five lines of ten or one line of fifty?

I'm working on a probability question with mean and variance. Let's say that I have two banks. They are identical in every way, except that bank A has five lines with ten people each and bank B ...
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14 views

Collective name for properties including expectation, variance, divergences, etc

Is there an established name for the class of properties of random variables such as expectation, variance, higher moments, divergences (e.g. divergence). These are properties of one or several ...
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30 views

Why do we divide by the standard error when we evaluate sample statistics?

I'm reading Experimental Design and Data Analysis for Biologists by Professor Quinn and Profesoor Keough and, on page 20, they write - We can use the methods just described to reliably determine ...
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How much can I parse the results of an A/B test without breaking the assumptions in the test?

Let's say I run an A/B experiment on 1,000 rats. I give one group regular food and the other a special type of food that should help their cognition and I compare their performance at mazes. Imagine ...
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25 views

Probability of event independent of random variables [closed]

Let {$X_n$} be a sequence of independent random variables and for each $n$, the event $A$ is independent of $X_1$, $X_2$,..., $X_n$. Show that $P(A)$ is $0$ or $1$.
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85 views

Using R to calculate probability of rolling 2 numbers in 20 throws [closed]

We have an 8-sided die. Throwing it 20 times, what is the probability of rolling a number larger than 6 at least 12 times and at most 16 times? If you know of an efficient program to write so that I ...
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1answer
25 views

How is the formula for the entropy of the lognormal distribution derived?

Wikipedia gives the entropy of the lognormal distribution in nats as $$\mu + \frac{1}{2} \ln(2\pi e \sigma^2)$$ Can anyone point me to a derivation of this formula?
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Why Kullback-Leilbler divergence is a better metric for measuring distance between two probability distributions than squared error?

I know that KL-divergence is a metric that is more suitable when we want to measure the distance between numbers which a probability form. However, I am still confused what is the benefit of using KL-...
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How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$

I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's. If I were to try to approximate ...
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1answer
24 views

“At least” approximation calculation

I have a vector of different probabilities to get 1, for example probs = [0.1, 0.5, 0.2, 0.9, 0.25, 0.55] I have to calculate the probability of having at least ...
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1answer
27 views

Expected number of days?

You’re drawing from a random variable that is normally distributed $X \sim \text{N}(0,1)$, once per day. What is the expected number of days that it takes to draw a value that’s higher that two? ...
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148 views

Why aren't “error in X” models more widely used?

When we calculate the standard error of a regression coefficient, we do not account for the randomness in the design matrix $X$. In OLS for instance, we calculate $\text{var}(\hat{\beta})$ as $\text{...
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1answer
53 views

Is a Bayesian posterior kind of like the marginal distribution of a frequentist estimator?

I've been thinking a lot about the relationships between various concepts like hypothesis testing, posterior distributions, and estimators. If I understand correctly, a frequentist estimator $\hat\...
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1answer
39 views

Force sum of random varables to equal to 1 [duplicate]

Suppose I have 3 random variables, $X1, X2,X3$. Define $Z$ as: $Z=X1+X2+X3$ I want to force $Z$ to equal 1 for every "realization" of $X1,X2,X3$ ($X_i \sim Beta(a_i,b_i))$. As an example, let $X_i$ ...
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2answers
45 views

How to add probabilities of a bad day or weather?

Sorry if this is too basic a question, or the wrong place to post. I am planning a road trip, and I'm trying to optimize the date I leave based on historically average weather for all the states I'll ...
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2answers
109 views

What is the probability of having a pair of doubles when throwing dice?

What is the probability of having a double (two dice will show the same number) when we throw 5 dice (all dice have 6 sides)? My calculations are: ...
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1answer
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Looking for a name for theory of a “balance of fair probability” (example provided) [duplicate]

I'm looking for a name of topic(s), theories which describe the situation provided below: Let's say we have a 6-sided fair dice. And after 6000 throws we have the following results: Digit 1: 1145 ...
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4answers
506 views

Expectation over a max operation

Let $X \in \mathbb{R}_{\geq 0}$ be a "non-negative" random variable and $c$ is a "given" strictly positive number. I wonder if the following inequality holds: $$ E[\max\{X,c\}] \leq \max\{E[X],c\}, $$ ...
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67 views

what is difference between $95 \%$ CI of mean and 95% pdf of normal distribution?

We took sample mean $\mu = 14$, and $\sigma = 0.45$. Calculate required area of normal distribution We will apply $\int_{13.19}^{15} f(x) \ dx = 95\%$. 95 % CI of mean But if we took sample size ...
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Predicting elections: Probability distribution from betting odds and dependence

Suppose I want to find the percentage of votes each party will get in elections. I have following odds from betting companies. My first thought is to find the probability for each party achieving a ...
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2answers
25 views

Can independent/dependent events be thought of graphically?

My understanding is that events are subsets of the total outcomes in a sample space. So if two events are mutually exclusive, then they (the sets) do not overlap in the sample space. This can be seen ...
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48 views

What are the main approaches to the foundation of statistics without probability

The frequentist, likelihood and, to an even greater extent, Bayesian approaches to statistics are all based on probability. Without probability, it seems difficult to use a data sample ("seen" cases), ...
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sum rule in conditional probability

P and S are the common cause of c. If P(C=true| P,S ) is given , can I introduce S to P(C|P) as P(C= true|P= true)= P(C=true| P =true , S= true)* P(P=true ,S=true )+ P(C= true | P=true ,S =false )*P(P=...
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How to eliminate variable given conditional probabilities

$P$ and $S$ are the common cause of $c$. If $P(C=true| P,S )$ is given as the table below, and $P(S=true) =0.3$, $P(P=true) =0.9$ how can I eliminate $S$ and calculate $P(C=true | P=true )$ and $P(C= ...
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1answer
18 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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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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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
29 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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1answer
22 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
42 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
117 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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31 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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25 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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0answers
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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0answers
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
25 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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23 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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0answers
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
48 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
32 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
71 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
207 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, ...