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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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Confusing multiple probability question

I'm trying something on time-series. I calculate the probability of a price movement in any given days. In this particular example here is what i have found: ...
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7 views

Standard variance of bivariate normal distribution plus normal distribution

Problem: $W = -27 + 0.3X + 0.45Y + E$ The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
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Probability of recurrence of the same two numbers in two consecutive lottery drawings

Given lottery game 6 out of 49, I would like to know how high the probability of recurrence of the same two numbers (i.e. 6 and 12) in two consecutive lottery drawings is. In this Wikipedia article I ...
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1answer
18 views

How can we estimate (conditional) probabilities from a dataset?

Given three random variables $X$, $Y$ and $Z$, how can we estimate $P(X)$, $P(X\mid Y)$ and determine whether $P(X \mid Y, Z) = P(X \mid Z)$ from a dataset (of e.g. $N$ observations) which contains ...
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How to prove that $X_t=\int^t_0f(u)dW_u$ and $X_t-X_s$ are independent?

Let $X_t=\int^t_0f(u)dW_u$ for a deterministic function $f$ and $W_t$ is a brownian motion. How can I compute $E[\exp(\lambda_1 X_s + \lambda_2(X_t-X_s))]$ and prove that $X_s$ and $X_t-X_s$ are ...
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9 views

Change in function value between correlated variables

Suppose i have dependent random variables $X_1,X_2,\ldots,X_n$ and a function $z = f(X_1,X_2,\ldots,X_n)$ and suppose i want to perturb one of the features $X_1$ by $\Delta X_1$ (i'm happy to use a ...
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17 views

Predicted Probabilities in Multinomial Probit Model

I am trying to compute predicted probabilities from a multinomial probit model. Researching the internet has given me two ways how this can be done. The first one is basically just applying the ...
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21 views

Joint distribution function

Suppose that we have two indpendent random variables $X, Y$ with a joint probability density function $f(x,y)=1$, $-y<x<y$, $0<y<1$ How can I calculate the cumulative joint probability ...
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18 views

Probability you end up at the origin after taking 2n steps?

Starting at the origin on the line we take a step of unit to the left or to the right with probability 1/2. We do this repeatedly with independent steps. If we take 2n steps, what is the probability ...
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1answer
13 views

Probability Question with k branches

Suppose my knowledge/ignorance of the number of branches of a certain store is given by the following probability law: P(k branches) = (1 − p)p^k where 0 < p < 1 and k = 0, 1, 2, 3, . . . If I ...
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2answers
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Finding the probability

Let $E,F$ and $G$ be three events such that the events $E$ and $F$ are mutually exclusive, $P(E\cup F)=1$, $P(E\cap G)=1/4$ and $P(G)=7/12$. Then $P(F\cap G)=?$ My attempt: Since $P(E\cup F)=1$. It ...
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1answer
29 views

Variance of sum of dependent random variables

Can you guys help me prove the following: $$ Var[\frac{1}{m}\sum_{i=1}^my_i]=\frac{1}{m}(1-\rho)\sigma^2+\rho\sigma^2 $$ where the sampled predictions ($y_is$) have variance $\sigma^2$ and ...
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0answers
13 views

Calculating the probability of exceeding a given return

I have calculated the return of an investment with cost and benefit data as 80% with a 4.6% coefficient of variation. However, I need to carry the analysis further by calculation the probability of ...
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1answer
35 views

Variance of linear combination of Normal distributions

A company that develops software received an order for a service to be performed within a week and, in order to decide on the profile of the team of programmers to be used, it should take into account ...
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0answers
17 views

Bayesian analysis: comparison of marginal probability distributions

Is it valid to compare mariginal probability distributions from separate Bayesian analyses to infer which scenario is most likely? Specifically, in phylogenetic (evolutionary) analysis, if I ...
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2answers
28 views

How can I calculate the probability of a increasingly likely positive outcome?

I'm not even sure I'm phrasing the question properly, please let me know if there is any standard terms around this type of problem. I'm trying to find an average number of attempts it would take to ...
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1answer
28 views

First passage time distribution via Monte Carlo simulation

The problem: I want to assess the first passage time distribution via Monte Carlo Simulation, where the first passage time is defined as: $$\tau=inf\{t: X_t > l\}$$ where $l$ is the barrier ...
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flight delay prediction [on hold]

Suppose that there is an airport and we have the average of flight delays for every day of the first month of 2018. Lets say the first day had an average flight delay of 2000 seconds and the second ...
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2answers
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Conditional Probability and Expectation for Poisson Process

To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $E(...
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1answer
22 views

Combinatorics with colored balls [on hold]

Seven blue and four red balls are to be arranged in order. How many ways can this be done if (1) The blue balls are distinguishable (e.g. numbered) as are the red balls. (2) Blue balls are ...
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1answer
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Which metric to use in an ordering problem? auPR / ROC / Lift?

I need to order Users from most likely to perform a binary action X in the next n days, to ...
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Expected Value Non-Central F Distribution

If $X$ is $\chi^{2}(5,\lambda)$ and $Y$, independent of $X$, is $\chi^{2}(4)$ then what is $$ E\left[\frac{X}{Y} + \frac{Y}{X}\right] = E\left[\frac{X}{Y}\right] + E\left[\frac{Y}{X}\right] $$ $\...
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Likelihood is “proportional to a probability”. Which one? [duplicate]

In various places (see quotes below) it says that the likelihood is "proportional to a probablility". Which probability is it proportional to? In the context of Bayes theorem, it is not proportional ...
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2answers
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Why use histogram to illustrated probability distribution

Forgive me I am a newbie of random variables. I saw a lot of course which introduce the Discrete Random Variables which always be illustrated with a histogram or ...
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1answer
38 views

Random Variables and Probability

So I encountered this problem while I was studying for exam. However, I cannot wrap my head around the solution that the answer key provided. The problem goes like this: Bob watches cars that pass ...
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1answer
14 views

Probability that one random variable using the Beta Distribution being greater than another, bounded intervals

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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0answers
15 views

Linear Regression in my time series class [on hold]

I need help on creating sas code for a linear regression for electric consumption versus time in years from april 1997 to april 2018. A sample of my data is shown below in the images. I am using SAS. ...
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0answers
27 views

Linear Regression (Time series Analysis) [on hold]

How do I do a linear regression of electrical consumption against year (time) from the electrical data set for the years March 1997- March 2018. A sample of my data is shown below in the images. I am ...
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1answer
27 views

What does it mean to “interpret the sigmoid $\sigma(\theta^Tx)$ as a probability”? [duplicate]

In Goodfellow's Deep learning text, it is written Is this way of defining a probability $p(y=1| x;\theta)$ even legal? Recall the definition of a probability given a random variable where $p_X$ is ...
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2answers
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Likelihood is not “proportional to” a single probability density?

In various places it says that the likelihood (e.g. in the Bayes formula) is "proportional to a probablility". For example https://alexanderetz.com/2015/04/15/understanding-bayes-a-look-at-the-...
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Notation for the assumed probabilistic model of our data

The question of best practices for probability notation is a common one on this forum, e.g.here, here, here, and elsewhere, e.g. here, here, here. Despite this, I still have unresolved questions. As ...
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Simplifying a covariance expression

Let $I=\begin{pmatrix}I_1\\\vdots\\ I_n \end{pmatrix}$ be a random vector, and $\Omega$ and $\Omega_I$ two random variables. I am trying to simplify the following equation (which worth $\frac{\rho_{I\...
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when are probability laws for events different from those of RVs? [on hold]

I had seen rules such as $P(A\cap B) = P(A|B) P(B)$, and similar laws such as $P(a,b) = P(a|b)p(b)$. In fact, I had generally disregarded the difference between these until now, incorrectly thinking ...
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What is the expectation of Poisson arrival times? [on hold]

I am interested in the expected wait time (till time $t$) of Poisson arrivals, $$\frac{\sum^{N(t)}_{i=1}{(t-t_i)}}{N(t)}$$ where $t_i$ is the arrival time of customer $i$, $i\in[1,N(t)]$. It is ...
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1answer
67 views

Interpretation of the rate parameter of a Gamma distribution

I am toying with mixture models, especially in a bayesian context and the Gamma (or the inverse Gamma) distribution appears quite often. For example, inverse Gamma is used as a conjugate prior for the ...
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2answers
25 views

Estimating Differential entropy from unbiased samples of a probability distribution

If I can get N unbiased samples $x$ from $p(x)$ how can I approximate the Differential entropy: $$H(X) = -\displaystyle\int_{x} p(x)\log p(x) dx$$ I'm not very knowledgeable in statistics so I'm not ...
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PDF of $\frac{(X_1-X_2)^{2}}{2}$ where X1 and X2 are independent standard normal [on hold]

I'm trying to find the PDF of $$\frac{(X_1-X_2)^{2}}{2}$$ where X1 and X2 are independent standard normal. Any hints on how to proceed with this?
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0answers
11 views

Goodness-of-fit test when the sample space is monotonically increasing

I came across this paper which develops a GoF test on data drawn from a circular sample space (i.e. it has cyclical support). I am now wondering if there is a parallel to monotonic supports (or if ...
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Questions on Notation for PMF and Expectation

Before diving into the Stanford CS229 Machine Learning notes online, I decided to go through the course's notes on probability review and had a few questions. In section 2.2, it states A ...
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Is f(x) = 1/n for x = 0,1,2,…,n a valid PMF? (n>0 an integer) [closed]

Is f(x) = 1/n for x = 0,1,2,...,n a valid PMF? (n>0 an integer) Intuitively thinking, the summation of the function over the range equals to (n+1)/n which is greater than 1 and so the function isn't ...
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0answers
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Probability of userbase recieving special page

Say I have an app. When the app is launched there is a process behind the scenes the determines which of the two outputs the user receives, default or special. The default output displays the app ...
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1answer
44 views

Proof of probabilities that may not be independent

I am given the problem: Given $P(A) = \frac{3}{4} $, $P(B) = \frac{3}{8} $, show that: a) $P(A or B) > \frac{3}{4} $. b) $\frac{1}{8} < P(A and B) < \frac{3}{8} $. The problem does not ...
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4answers
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How to build a Bayesian Model to estimate the probability distribution of the parameters given the output?

I'm currently facing a new type of problem, and i have no idea how to solve it, so any suggestion will be really appreciated ! The problem is the following: I have a matrix of temperatures, depending ...
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1answer
26 views

differences between conditional probability and dependency

Sometimes, I read articles about conditional probabilities and other articles about conditional dependency. My question what is the main differences between them? For example, "https://en.wikipedia....
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0answers
26 views

Independent Study Statistics/Probability Grad Level [duplicate]

I am trying to decide on topics for my independent study this semester. I am a Pre-Doctoral Mathematics student, so looking for a more math based text rather than engineering based (which I have found ...
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0answers
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Co-occurence in a population

Let's say I have a population of size N that has traits a-z. Some people/samples might have just a,n,x, etc... P(a), ..., P(z) is known. # of people with i & j in the population is known for all ...
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0answers
8 views

Series Winner Probability

Suppose we have Series 1 and Series 2, and four teams $A,B,C,D$. $A$ and $B$ play in Series 1 and $C$ and $D$ play in Series 2. In each series, the first team to win 3 games wins the series. The home ...
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Probability Winning Game [duplicate]

Suppose you are tied $1-1$ with Bob. To win the game, you have win greater than or equal to $5$ rounds and win at least $2$ more rounds than your opponent. The probability of winning a round is $40\%$ ...
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1answer
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Probabilistic user behavior markov models on web

I am considering the following probabilistic Markov model of actions of a user on the results page of a search engine. The user examines the first result, with a probability $A$ he is satisfied with ...
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Find the distribution function $F$ for $min_{1 \le i \le n}{X_i}$ [duplicate]

Given a random sample $X_1, X_2, ..., X_n$ where each $X_i$ has pdf: $$ f(x; \theta) = 3 \theta^3 x^{-4} $$ and $0 \lt \theta \le x \le \infty$. Show that the distribution function $F$ for $min_{1 \...