# Questions tagged [probability]

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

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### Can a random variable be expressed as a sum of deterministic and random variable?

Say we have a sequence of random variables $\{X_t:t\geq 0\}$ following an unknown stochastic process with distribution $X_t\sim N(\mu_X,\sigma_X^2)$. This idea came to me from the additive noise model....

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### What does “version” mean here?

In a paper I read about the following statement:
"Assumption 1. There is a version of $f(x)$ that is twice continuously differentiable"
Note that $f(x)=E(Y|x)$ is an unknown function to be estimated ...

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### Determining the power of the test in this question

The following problem is from Devore's Probability and Statistics for Engineering and the Sciences, 8th edition, exercise 8.1 question 33:
Reconsider the accompanying sample data on expense ratio(%...

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### R: Question about central limit theorem

Hello everyone :) can you help me please, I really don't understand my teacher's videos and it is the last part of our 20-pages work :O
In the question 1 they ask us to create a Poisson distribution ...

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### Here you roll seven dice,: A grand prize of $5,000 if you roll seven of a kind $500 if you roll six of a kind $50 if you roll f [closed]

mathematic.probability expected value

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### Is this even possible?

How is this possible:
If $P (Z|Y, X) = P(Z|Y)$ AND $P(X|Y,Z)= P(X|Y)$
How can these two be equal: $$P (Z|Y,X) = P(X|Y,Z)$$

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### How do I solve this question on poisson and exponential distribution? [closed]

The SureWin Brokerage Firm manages portfolios of common stocks. Its computers mon- itor stock prices and, when certain conditions arise, they issue buy or sell signals. These signals follow Poisson ...

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### Regression equation given a joint distribution

Let $X$ and $Y$ be two random variables with joint probability density function
$f(x,y)$ = $ \left\{
\begin{array}{ll}
1 & -y\leq x\leq y , & 0 < y <1\\
0 & ...

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### If $E(Y|X_1=x_1,X_2=x_2)=0$ and $Var(Y|X_1=x_1,X_2=x_2)=C$ for all possible $x_1,x_2$, then what is $E(Y|X_1=x_1)$ and $Var(Y|X_1=x_1)$?

Given random variables $Y,X_1,X_2$, if $E(Y|X_1=x_1,X_2=x_2)=0$ and $Var(Y|X_1=x_1,X_2=x_2)=C$ (where $C$ is a constant) for all possible combinations of $x_1$ and $x_2$, then is $E(Y|X_1=x_1) = 0$ ...

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### HMM backward probability question(please help)

Hello my first question here
I am was learning nlp, and recently was researching about HMM. Just to make sure I understand it correctly so we have to make two assumptions to simplify everything for ...

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### If A and B are independent, can P(C | A, B) be expressed only in terms of P(A), P(B), P(C | A), and P(C | B)?

Conditional probability question.
Let's say I have...
three random variables: A, B, C
<...

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### Can I use z-scores with an exponential distribution? Or is there another test statistic for these types of distributions?

I have an exponential distribution for a population. $\theta = \mu = \sigma$ is known. Sample size $n$ is known. I need to find $"š(š<šĀÆ<š)"$ for a random sample with size $n$.
I think I am ...

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### Distinguishable or Indistinguishable: Occupancy Problem

Problem Statement: Contracts for two construction jobs are randomly assigned to one or more of three firms, $A, B,$ and $C.$ Let $Y_A$ denote the number of contracts assigned to firm $A,$ and $Y_B$ ...

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### Is the mean of two symmetric, independent random variables also symmetric?

Does a proof exist that the mean of two symmetric, independent random variables is also symmetric? Or is the conjecture false?
I would be interested to learn about references to this question, or ...

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### Extending normalized probability generating function (pgf) in branching process

My question is in the context of branching processes and simply how to extend a normalized negative binomial probability generating function (pgf) from $\frac{1}{y}[G(s)]^y$ to $\frac{n}{y}[G(s)]^y$ ...

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### Proving Two Standard Normal Variables Are Uncorrelated

Let $X \sim N(0,1)$, a random variable $U$ does not depend on $X$, and $P(U = 1) = P(U = -1) = 1/2$. Finally $Y = UX$. Prove that $X$ and $Y$ are uncorrelated.
$Y$ should also be standard normal due ...

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### Success rates with different sample sizes

The same item is on sale in 3 different shops in town:
Site "A", they find 14% defective items out of a sample of 100, being the total number of items in stock 1000.
Site "B" they find 12.5% ...

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### Difference of two KL-divergence

The Kullback-Leibler (KL) divergence between two distributions $P$ and $Q$ is defined as $$\mbox{KL}(P \| Q) = \mathbb{E}_P\left[\ln \frac{\mbox{d}P}{\mbox{d}Q}\right].$$
My question is that suppose ...

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### Number of permutations with fixed first and last places

If I have 3 males and 2 females lining up at a grocery store, how many different ways can they line up if the first person in line is female and the last person in line is male?
Is it as simple as ...

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### Card Probability Question: No Replacement, dealing cards until getting a heart

Cards are dealt without replacement until a heart appears.
I had to find the P that the first heart is on the 4th, so I did: Prob of "not heart"on:
1st = 39/52, 2nd = 38/51, 3rd = 37/50, so Prob of ...

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### How to compare two Gaussian distributions

If I have two Gaussian distribution with the same $\sigma$ but different $\mu_x$ and $\mu_y$. How to calculate the $P(x>y\ |\ \mu_x\leq\mu_y)$? I think it's a type 1 error when I set the event "$\...

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### Restricting Binomial Success Rates in simulation

Suppose I am generating 10000 data points with probability P out of N total trials, but I know that P x N can never be below M, where M < N. How would I go about restricting my simulation to never ...

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### Gradient map based on probability of incident

I have a binary classification model that outputs the probability of an instance belonging to the positive or negative class. I already have the probability threshold by which an instance will be ...

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### Hypothesis Test on the Difference between two random vectors

Each of my vectors consists of beta estimates for two separate models of the same data and the same number of explanatory variables. The question is asking whether the difference between these two ...

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### chisq.test in R isn't giving correct X-squared [closed]

probs <- dpois(0:3, lambda = 1)
comp <- 1-sum(probs)
O <- c(4,3,2,1,0)
E <- 10*c(probs,comp)
chisq.test(O, E)
Running this code ...

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### Gaussian mixture models for image matrix not determining E step

I want to calculate responsibility for each of the data points, for the given MU, SIGMA and PI.
...

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### How do we obtain the probability density of a truncated regression with an upper and lower bound

I know my density for $y$ is supposed to be something of this form $$g(y|x_{i},t)=\frac{f(y|x'\beta, \sigma^{2})}{F(t|x' \beta' \sigma^{2}}$$ where the numerator is the density of the normal ...

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### Prove $\lim_{n\to\infty} X_{n} = \lim_{n\to\infty} Y_{n}$ implies that $\lim_{n\to\infty} E[X_{n}] =\lim_{n\to\infty} E[Y_{n}]$

Prove $\lim_{n\to\infty} X_{n} = \lim_{n\to\infty} Y_{n}$ implies that $\lim_{n\to\infty} E[X_{n}] =\lim_{n\to\infty} E[Y_{n}]$ given $X_{n}, Y_{n}$ are increasing sequences of positive integrable ...

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### Probability of winning a competition K games best of series of N games

Consider a 'best of 5' series in sports/competition where the first team to win 3 games wins the series. So N=5, K=3. Where probability w = p(team A winning game) ...

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### Understanding notation in Bias-Variance decomposition in Elements of Statistical Learning

I'm going through Elements of Statistical Learning and I'm having a bit of trouble understanding this bit of notation from Chapter 2 (this example is (2.27))
$$EPE(x_0) = E_{y_o|x_o}E_T(y_0 - \hat{y}...

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### Expected value for European Roulette Probability

below is the query to calculate expected value?
Probabilities P(x)= 1/37 = 0.0270 P'(x)=1-1/37=0.9729
Expected value=100 * 0.0270-0.9729*100 for betting 100 $ on number 5,
What could be wrong ...

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### Mathematically, 1 in 3 and 10 in 30 are equal. What about in probabilities? [closed]

Assume I am in a draw for a prize. There are 3 rubber balls, 2 red ones (loser) and 1 green ball (winner). If the green ball is drawn, I win.
Now let's have a second draw (simultenously). It has 20 ...

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### Error : data must be a numeric vector of length greater than 1 [closed]

Please help me to solve this error, thank you.
pWeiNorLL <- function(q,mu,sigma,c,gamma) 1-exp(-(pnorm(q,mean=mu,sd=sigma)/(gamma*(1-pnorm(q,mean=mu,sd=sigma))))^c)
dWeiNorLL <- function(...

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### Expecation of Linear Regression Coefficients

Let the entity ${\widehat{\boldsymbol\beta}}$ be a linear estimator (not necessarily the least squares estimator) of the true coefficient ${{\boldsymbol\beta}}$ in the regression of š² on š. In this ...

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### Probability of observing a number of special balls from a larger set

Suppose there is a collection of $n$ balls of which $m$ are special. What is the probability of drawing $k$ special balls, when $p$ balls are drawn?
To give it a try I considered the following ...

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### Some questions regarding independence requirements in the conditional likelihood

For iid random variables $Y_1,\dots,Y_n$ from a random sample that are mutually independent with pdf $f(y|\theta)$, my book defines the likelihood as $L(y_1,\dots,y_n|\theta)= f(y_1,\dots,y_n|\theta) =...

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### Given random variables $X,Y,Z$, under what conditions is $P(Y|X)=P(Y|X,Z)$?

From this link, where the statement is given for events and not random variables, I gather that for random variables $X,Y,Z$, $P(Y|X)=P(Y|X,Z)$ only if $P(Y,Z|X)=P(Y|X)P(Z|X)$? Does this imply that $Y$...

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### How to simulate predicted probabilities

Can you help me out with the following brain twister?
I have a prediction model to estimate the probability (p) of a sale for each potential customer. On average, p is 0.003. (So there is approx. one ...

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### Bayes' Theorem - is the probability quantifiable in this case?

Question 4.49 in Newbold (8. ed)
A company receives large shipments of parts from two sources. Seventy percent of the shipments come from a supplier whose shipments typically contain $10\%$ ...

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### I need to construct function that gives out data of specific pdf function and draw histogram of it

So, the given density function is: $\theta x ^{\theta - 1}, \ \ if \ \ 0<x<1 \text{ and } 0 \ \ \text{ otherwise}$. $\theta$ is given as 2. I also need to draw a line that corresponds the ...

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### Use of probabilistic sensitivity analysis in determining incidence over time

I'm plotting incidence (y, per 10,000 population) over time (x, years) using results from a simulation model. And seeing when the incidence reaches below a certain incidence target (1/10,000). It was ...

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### Quantifying Co-occurrence

I have a problem where I need to quantify the co-occurrence of two events. I've been using a working metric I'm calling "normalized co-occurrence rate". It's the proportion of events that occur ...

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### How to compute co-occurence probability from a co-occurence matrix?

I have a co-occurence matrix in this format:
...

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### Shannon Information | Understanding from a Microstate Perspective

So Shannon's information is a way to quantify "distinct knowledge" by means of combination of microstates. So say 1 bit of information in binary system conveys 2 sets of information due to two ...

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### If an event is mutually exclusive find $P(A' \cap B')$

So the question says that you have A and B event they are mutually exclusive so find
$P(A' \cap B')$ I have solved it by :
$$P((A \cup B)') = 1 - P(A \cup B) $$
But what is the effect of being ...

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### What distribution does the mean of a random sample from a Uniform distribution follow?

For example, let $X_1,\cdots,X_n$ be a random sample from $f(x|\theta)=1,\theta-1/2 < x < \theta +1/2$.
Clearly, $X_i \sim U(\theta-1/2 , \theta +1/2)$. Some intuition would suggest that $\bar{X}...

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### The probability of an observation belonging to one or other sample

Assume I asked 10 children and 10 adults whether they like my cake. 9 kids did and 1 didn't, while for the adults it was the reverse. Using these data, can I say that any random person who likes my ...

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### Conditional Probability for Coin Flip Data

I am reading the Tutorial of Fisher Information and it is mentioned that the say we have a random variable $X^n$ where $n$ refers to number of tosses so the RV $X^2$ could be {$1,1$} or {$0,1$} then ...

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### How pvalue is calculated internally here ? And what is the data point for which it is calculating pvalue

I got a into a question while going through poisson distribution.
Here it is:---
A nuclear pump failed 5 times out of 94.32 days.
Give a 95% confidence interval (CI) for the failure rate per day
So ...

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### Conditional expectation for maximum function

I have a discrete-time Markov chain queuing problem.
Packets (computer packets, that is) arrive in the intervals. $A_n$ denotes the number of arrivals in the interval $(n - 1, n)$, where $n \ge 1$, ...