# Tagged Questions

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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### Why does $n \choose r$ where $r = 1,n$ track $2^n$?

I bashed together a clunky ruby script to find the sum total of $n \choose r$ where $r = 1,n$ I wanted to determine how many lines of output I could expect from a script that produces all possible ...
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### Ask for a question about independence of random variable from an event

Consider two independent tosses of a fair coin. Let random variable X take the value 0 if the first toss is a head and take the value 1 if the first toss is a tail. Let A be the event that the number ...
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### Probability interview question

Suppose we have three positive integers $A, B, C$. We randomly choose an integer $a$ in the range $[0,A]$ and an integer $b$ in the range $[0,B]$. Find the probability that $a + b\leq C$. I am unable ...
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### Proof that the sum does not depend on enumeration …

I've come across the following basic lemma in a basic probability book, and I can't seem to understand why the provided argument is enough to prove it. Lemma: Let $I$ be a countably infinite set and ...
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### Continuous and measurable in each variable $\implies$ product measurable?

Consider a metric space $A$ with a metric $d$, and consider the measurable space $(A,\mathcal{B}(A))$ with the Borel $\sigma$-algebra generated by $d$-open sets. Let $(\Omega,\mathcal{F})$ be a ...
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### The law of large numbers - limits of $\max$ vs $\max$ of a limit.

Assume that $X_{1,1}, \dots , X_{1,n}, X_{2,1},\dots, X_{2,n}, \dots ,X_{n,1}, \dots , X_{n,n}$ are i.i.d. random variables, and that $\mathbb EX_{i,j}$ exists and is finite. From the strong law of ...
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### Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
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### computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
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### Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
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### Weak convergence of a double sequence of random variables

Consider two sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Let's assume $X_n\xrightarrow{D} X$, $Y_n \xrightarrow{D} Y$, and $\{X_n\}$ and $\{Y_n\}$ are independent of each other. It is well-...
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### Probability & statistics

Q: In a given town, $270$ days of the year have a winter-like weather and the other $95$ days have a summer-like weather. On a winter-like day the probability of a sunny day is $0.3$. On a summer-like ...
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### Central Limit Theorem and understanding mean for a single object

The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students ...
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### How to solve “ways of seating around a circular table”

Recently I asked a question about seating, here it is again: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five ...
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### Type I error in Normal distributions

Let $X_1,\dots , X_n \stackrel{iid}{\sim} N(\mu, \sigma^2 = 4)$ Test $H_0: \mu = 10$ vs $H_1: \mu > 10$ take a random sample of $n=16$ and reject $H_0$ if $\bar{x}>14$ Find $\alpha$ the type I ...
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### Type I and type II errors

Let $X \sim uniform(0,\theta)$ we are testing $H_0: \theta = 1$ vs $H_1: \theta >1$ If we know that we reject $H_0$ if $X>0.9$ (1) find $\alpha$, the type I error (2)Suppose that $\theta=1.1$. ...
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### getting intuition about a fact on probability

Let $X: \Omega \to \mathbb{R}$ be a random variable.let $L^1$ be the set of all random variables which have finite expectation. My books says that if $X^2 \in L^1$, then $|X| \in L^1$. My question is:...
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