# 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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### 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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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 $"𝑃(𝑎<𝑋¯<𝑏)"$ 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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### 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 ...
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 $n \ge 1$, ...