Questions tagged [probability]

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

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1answer
22 views

Which is the event, sample and population, notation confusion

I am really struggling with the notation for this. Below is the slide material followed by what I think it means/where I get stuck. "Some unknown real world quantity $\Theta$ takes values in $\Omega$....
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47 views

Best way to estimate the probabilities of a random variable

I have some confusion about estimating the probability of a particular value of a random variable. For simplicity, consider the case of a coin and the random variable being $X = \{H,T\}$, where $T$ is ...
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3answers
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How to calculate a confidence interval for a series of Bernoulli Trials?

I have to test if a event have a p probability of happening. I can run this event as much times I like (given it can be run by a computer). So I was searching a way ...
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1answer
38 views

Build model for daily probability of meeting a certain end of period goal

I was hoping for some consultation with how to go about the following: To give context, I work for an agency that manages advertisements on social media for general motors - specifically their car ...
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14 views

In the fully supervised case, provided we have contingency matrices, is Bayesian inference the optimal method?

BACKGROUND Imagine that we have contingency matrices, i.e., counts or frequencies, linking the features (say, columns) and targets (rows). One could then compute the posterior probabilities, i.e., ...
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1answer
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Need help to understand Feller's statement “whenever $r$th moment exists so do all preceding moments”

I am reading the book of Feller called "An Introduction to Probability Theory and Its Applications, Vol I" (third edition, page 227) and am stuck at the moment he explains the notion of variance of a ...
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4answers
1k views

Why is the concept of the Null hypothesis associated with the student's t distribution?

There are dozens of continuous probability distributions like Gaussian (normal), Variance-gamma, Holtsmark, etc. Yet, the concept of the Null hypothesis is basically associated with Student's t-...
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1answer
43 views

Time series forecast with probability

I have historical data for a particular metric for each month for the last 3 years for different categories. The metric is a percentage and its heavily skewed towards 1 with more than 75% of values ...
2
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1answer
34 views

Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
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25 views

Difference between tight and uniformly tight random variables?

This wikipedia page implicitly says that “tight” and “uniformly tight” random variables refers to the same concept. I find this somewhat surprising. Are there contexts in which a distinction is made ...
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22 views

Quantile (Inverse Cumulative Density) Function for Hypergeometric Distribution

The hypergeometric distribution arises from sampling without replacement. The similar binomial sampling distribution assumes replacement. Hypergeometric distributions are commonly used in quality ...
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1answer
136 views

Probability and Counting

Suppose You are ordering two pizzas.A pizza can be small,medium,large or extra large,with any combination of 8 possible toppings( getting no toppings is allowed, as is getting all 8) How many ...
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Uniform prior 3 sided dice marginal probability equation [duplicate]

Given a 3 sided dice with a uniform prior. What is the probability of observing ordered data $D = \{n_1, n_2, n_3\}$. Where $n_1$ is the number of observed 1s. Denoting the bias on the dice by $\...
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22 views

Estimate run time given widget has run a given time without failure

Given the following distribution of the fraction of widgets running over time, how do I calculate the expected run life of a widget (d_f days) given that it has already run d_c days? My widget ...
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0answers
38 views

Characterizing a distribution

I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year. If I have some data in the form ...
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13 views

Conditional model with correlation between estimations

I am trying to estimate Click-Through-Rate (CTR) given the following two models: $$P(Click|Visible)$$ $$P(Visible)$$ The output is: $$P(Click) = P(Click|Visible)*P(Visible)$$ Unfortunately, $$P(...
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Tail of the CDF of noncentral chi-squared RV

The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}...
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2answers
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Probability that all cards have been drawn [duplicate]

Let's say, I have a deck of $m$ cards from which I will draw $n$ cards at random, watch them and put them back in the deck. I want to know; what is the change after $k$ draws, that I have seen all ...
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Gauss's Original Gaussian Distribution Proof Help

I am refering to the proof of the guassian from the famous document: https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf I have attached the pictures below of the ...
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1answer
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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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2answers
58 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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38 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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29 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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1answer
34 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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1answer
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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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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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31 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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8 views

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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1answer
27 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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2answers
87 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
28 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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1answer
161 views

Why is Kullback-Leilbler divergence a better metric for measuring distance between two probability distributions than squared error? [duplicate]

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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35 views

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
25 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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2answers
161 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
56 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
41 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
46 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
112 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
27 views

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
534 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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1answer
68 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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0answers
17 views

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
28 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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0answers
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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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1answer
35 views

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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1answer
52 views

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= ...