Moments or Mean of transformed Weibull Distribution.
Given the PDF and CDF of the Weibull distribution, denoted as $f_X(x)$ and $F_X(x)$ respectively, the range is $x \geq 0$.Also, clearly that the mean, $E(x)$ would be $\int^{\infty}_{0}xf_X(x)...
View ArticleExpected value of game assuming following optimal strategy
We play a game. The pot starts at $\$0$. On every turn, you flip a fair coin. If you flip heads, I add $\$100$ to the pot. If you flip tails, I take all of the money out of the pot, and you are...
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Understanding Standard variance value vs Expectation value scatter plot graph
I am dealing with the following scatter plot graph of expected and standard variance values of different distances for 100 randomly chosen pairs of intervals:Here $d_I, d_{Hm}, d_E$ and $d^H_{L1}$...
View ArticleLimit of Expectation values involving exponential i.i.d random variables
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent and identically distributed (i.i.d) random variables with $\mathbb{E}[X_1] = 1$ and $\mathbb{V}[X_1] = 1$. Show that$$\lim_{n \rightarrow...
View ArticleExpected hitting time in at least one walk with multiple 1-side bounded...
Let $S_n^1,\dots,S^d_n$ be $d$ mutually independent simple one-dimensional random walks on $\mathbb{Z}$ with $x=S_0^1=\dots=S_0^d$. In each time step each of the $d$ random walks moves, independently...
View Article$\sum_{k=1}^{\infty} X_k < \infty \to \sum_{k=1}^{\infty} VarX_k < \infty$
Let $(X_k)$ be independent random variables, with $P(X_k \le M)=1$, i.e bounded by a constant M. Proe that if $\sum_{k=1}^{\infty} X_k < \infty$ and for $k \in \mathbb{N}$$EX_k = 0$...
View ArticleHow to find the origin of the formula...
According to Wikipedia, the expected value of a continuous random variable can be calculated by,$$\mathbb{E}[X]=\int_0^{\infty}(1-F(x))dx-\int_{-\infty}^0F(x)dx$$where $F(x)$ represents the cumulative...
View ArticleUpper expectation bound for generalized multiplication
Let $\{X_k\}_{k=1}^N$ be a sequence of positive independent random variables such that $\mathbb{E}[X_k] \leq 1$.Let $\phi:(0,\infty) \to J$ such that:$\phi$ strictly increasing and concave.$\phi(1) =...
View ArticleExpected hitting time in at least one walk with multiple 1-side bounded...
Let $S_n^1,\dots,S^d_n$ be $d$ mutually independent simple one-dimensional random walks on $\mathbb{Z}$ with $x=S_0^1=\dots=S_0^d$. In each time step each of the $d$ random walks moves, independently...
View ArticleExample of a general random variable with finite mean but infinite variance
Given a probability triple $(\Omega, \mathcal{F}, \mu)$ of Lebesgue measure $[0,1]$, find a random variable $X : \Omega \to \mathbb{R}$ such that the expected value $E(X)$ converges to a finite,...
View ArticleWhat's the expected value of a coin game in which I win $N$ or lose $3N/4$?
This should have a simple solution but I'm having a hard time trying to understand it. If I play a game in which I win $20$ dollars when a coin lands heads and lose $15$ when it lands tails, the...
View ArticleUseful approximations of undefined first moment
In the process of thinking about this problem, I had a question regarding a related idea.To summarize, I started by considering the expected number of times that an arithmetic brownian motion would...
View ArticleSum of a subset of i.i.d. Bernoulli r.v.s conditional on their total sum
Suppose we have i.i.d. $\xi_1,\dots,\xi_n\sim Bernoulli(p)$ and define $\tau=\sum_{i=1}^n\xi_i,\eta=\sum_{j=1}^{\tau}\xi_j$.Find $\mathbb{E}\eta$.Unlike in some of the similar problems solved here,...
View ArticleConditional expectation of family of random variables parametrized by a...
Suppose we have a random variable $X_s$ that explicitly depends on some number $s$, seen as a parameter for instance. Now, let $Y$ be a random variable such that for every $s$, $X_s$ and $Y$ are...
View ArticleThe converge of expectation value based on almost sure convergence
Here is the question:Let $\xi_n $ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P) $ such that $E \xi_n^2 \le c $ for some constant $c$. Assume that $\xi_n \to \xi $...
View ArticleExpected Value in Random Sampling with Replacement and Variable Probability
There are $n$ white balls in an urn. We draw one ball with replacement. If it is white, we paint it red before putting it back. Let $X$ be the random variable representing the number of red balls in...
View ArticleExpectation of random variable with a tail condition
Let $T$ be a real valued absolutely continous r.v. such that $\lim\limits_{t\to +\infty} t^a \mathbb{P}(T>t)=1$ for some $a<1$, is it true that its expected value must be infinite?I found the...
View ArticleI flip a coin 10 times. What is the expected number of pairs of consecutive...
For example, the sequences $THHTHHHTHH$ has 4 pairs of consecutive HH's.I question the approach of a solution in my textbook. It goes as such: There are $2^{10}=1024$ possible sequences of ten coin...
View ArticleWhy is shooting 3s better than shooting 2s, with a score-to-win?
I play a lot of low level pickup basketball. Out of curiosity, I wanted to simulate the difference in outcomes between shooting threes and shooting twos. Specifically, I was curious what things look...
View Article51 points are inside a square of side length 1. Prove that we can cover some...
The traditional answer uses the pigeonhole principle to solve this, but I attempted a different way and I can't figure out why my solution is incorrect. I attempt to show this by finding the expected...
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