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A square contains many random points. From each point, a disc grows until it...
A square lamina contains $n$ independent uniformly random points. At a given time, each point becomes the centre of a disc whose radius grows from $0$, at say $1$ cm per second, and stops growing when...
View Articlesequence of random variables converge to $X$ in probability with all...
I'm currently working on the following problem:Construct a sequence of random variables $X_n$ and $X$ such that $X_n \overset{\mathbb{P}}{\rightarrow} X$ and $\mathbb{E}[X_n] = 0$ for all $n$ but...
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A square contains many random points. From each point, a disc grows until it...
A unit square lamina contains $n$ independent uniformly random points. Each point is the centre of a disc whose perimeter touches the nearest neighboring point. Here is an example with $n=20$.In this...
View ArticleProve that $\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$ for i.i.d $X$ and $Y$
Let $X$ and $Y$ be two independent identically distributed random variables with finite expectation $\Bbb{E}(X) = \Bbb{E}(Y) < \infty$. Prove that$$\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$$I think that...
View ArticleDifferences of order statistics for symmetric random variables
Take a sequence of $n$ i.i.d. random variables symmetric around zero and with zero expectation:$$\eta_1,\eta_2,\dots, \eta_n.$$Use standard order statistic notation and consider$$\eta_{(1)}\leq...
View Articleexpected value of the product of two Bernoulli variables
I'm stuck on proving that the product of two Bernoulli variables has expected value equal to the difference between the probability that the variables are the same and the probability that the...
View ArticleInvalid Conditional Probability - Expectation of Die Rolls
Problem: Roll a fair 6−sided die until a 6 appears.Given that the first 6 occurs before the first 5, find the expected number of times the die was rolled.Note: A discussion on the solution already...
View ArticleExpectation of product of iid random variables and function
Suppose $X_1,X_2,\ldots,X_n$ are iid random variables with mean $0$. and let $Y_i=\sum_{k=1}^nX_k -X_i$. Clearly $X_i$ and $Y_i$ are independent. I want to find the expectation for $i \neq j$$$\mathbb...
View ArticleConditional expectation of length of path
We have $n$ houses, $n\geq 2$, that are set in a straight line, such that the distance between any two neighboring houses is $a\in (0, \infty)$. A resident from a randomly chosen house goes on a visit...
View ArticleProbability question: Expected number of trips for an alien to visit all $n$...
I came across problem which I cannot seem to solve when self studying probability. I was wondering if people can help me out.There are $n$ planets in an orbit. So, there are $n$ planets in acircle. An...
View ArticleUniform Distribution Game
Here is a probability game description:You generate a uniformly random number in the interval(0,1). You can generate additional random numbers as many times as you want for a fee of$0.02per generation....
View Articlewhat’s the standard deviation of this final value?
Interval I0= [0,1], the i^th interval is randomly chosen from (i-1)th interval with half of its length, at the end it converges to one point, what’s the standard deviation of this final value?e.g.,...
View ArticleExpected number of heads in a sequence with no two consecutive heads [duplicate]
Given a fair coin is tossed $n$ times and the resulting sequence contains no two consecutive heads, Let $E(n)$ be the expected number of heads, what is $\displaystyle \lim_{n\rightarrow\infty}...
View ArticleExpectation of new measure
Fix probability space $(\Omega, F, P)$, let $D$ be a nonnegative random variable with $E[D] = 1$. Define a new probability measure $Q$ on $(\Omega, F)$ by $$Q[A] := E[D 1_A]$$I have the following...
View ArticleCalculation of Expected Value for the Ratio of Variance to Three Times the...
As a doctoral student working on estimation, I encountered the following problem: I am seeking assistance in calculating the expected value of the ratio$E(\sigma^2/\3*mu),where $\sigma^2$ represents...
View ArticleExpectation of a power of CDF
If $X$ is a random variable with distribution function $F,$ then $\mathbb{E}[(F[X])^{-1/2}]$ can be computed by integration by parts, if $X$ has a continuous density $f$. What happens in the general...
View ArticleBirthday paradox - variance, parallelisation, simple proofs?
Suppose we sample uniformly random elements from a set of cardinality $n$, and save them in a table. We continue doing this process (each sampling is one step) until we get a collision. What is the...
View ArticleTaking the limit in Holder's inequality
I have a standard normal random variable $X\sim\mathcal{N}(0,1)$ and an event $E$ with $\mathbb{P}(E)=p$, and this event is about $X$ and some other variables. I am interested in upper-bounding the...
View ArticleExpected value of X^p, where X is drawn from a truncated normal distribution
I am trying to find a closed solution to find the expected value of $X^p$ with $p>0$, where $x$ is drawn from a truncated normal distribution cut on both sides, such that it is non-zero only in a...
View ArticleExpected sum of three dice if same value are cancelled
I came across with the problem of calculating the expected sum of two fair dice where if the two numbers are the same, the sum is zero. This can be done by simply summing all possible cases and then...
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