Expected value of two dependent Bernoulli random variables.
I'm in the process of finding $E(XY)$ where X and Y are non-independent Bernoulli random variables with both of them with probability of $\frac{n}{N}$I understand that $E(XY)=\Sigma\Sigma...
View ArticleUpper Bound for the Difference Between Expectations
Let us consider two random variables, $X_0 \sim \mathbb{P}_0$ and $X_1 \sim \mathbb{P}_1$. Is it possible to establish an upper bound to the difference between the expectations of $X_0$ and $X_1$ based...
View ArticleTwo Jokers problem
Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?This is an HMMT...
View ArticleFollow-up to Expected-Value Dice question
Problem: Given that the first 6 occurs before the first 5, what is the expected number of rolls of a fair six-sided dice it will take to roll a 6 for the first time and stop?This question was asked...
View ArticleCutting a rod in n parts [duplicate]
A rod of length 1 unit is cut into n pieces.Find the expected value of the smallest and largest piece.
View Articlethe expected value of the square of the distance of random two points on a...
Here is a problem from this year's MathCounts.Two points are randomly and independently chosen on the circumference of a circle of radius 6. What is the expected value of the square of the distance...
View ArticleWhat is the value of $\operatorname{Var}(S^2)$, when $S^2$ is given as sample...
At first, this looked like an easy task, starting with usual expression for variance, such as:$\operatorname{Var}(S^2)= E(S^4) - (E(S^2))^2$. Once again, herein I want to define $S^2=\frac{\sum_1^n...
View ArticlePower diffs: $\frac{1}{\gamma}\mathbb{E}[X^\gamma - X_*^\gamma] \leq 0...
I would like to show that the following implication is true for all $\gamma<1$ and $a$ in the $\mathbb{R}^d$ simplex $\Delta^d$:$$\frac{1}{\gamma}\mathbb{E}[(a^TX)^\gamma - ({a^*}^TX)^\gamma] \leq 0...
View ArticleNormalized hamming distance in probability
Let two functions $f,g :[n]\to \{0,1\},$ then we define $$\delta{(f,g)}=\frac{|\{i\in[n]:f(i)\neq g(i) \}|}{n}$$ is called normalized hamming distance.My teacher said me this $\delta{(f,g)}$ is...
View ArticleIs the expected value of a continuous random variable always defined if its...
It is well known that there are random variables for which the expected value is undefined, e.g., if a random variable has a Cauchy distribution. However, the Cauchy random variable is supported on the...
View ArticleMinimum squares with two different means?
$X∈U(0,θ)$ and $𝑌∈𝑈 ( 0 , 4𝜃 )$ . $𝑋$ and $𝑌$ are independent. We want to estimate 𝜃 using the Least Squares Method and we obtain the outcomes $𝑥 = 5.2$ and $𝑦 = 28.3$. Determine the Least Squares...
View ArticleEstimate clan size
The people in a country are partitioned into clans. In order to estimate the average size of a clan, a survey is conducted where $1000$ randomly selected people are asked to state the size of the clan...
View ArticleCalculating Expected Value for a Carnival Game
There is a carnival game of pure luck, and there are $2$ prizes. One is a jackpot with a $67/63000$ chance of happening, and a smaller prize with a chance of $25/216$ of happening. The player must pay...
View Article$\mathbb{E}[B^4(t)]$ with $B$= brownian motion
Can anyone help me to find:$\mathbb{E}[B^4(t)]$ where $B$ is a brownian motion?I thought using this density function:$f_{B_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-\frac{x^2}{2t}}$,but I don't know how to...
View ArticleSmoothness of expectation of a piecewise function
Suppose $f(x)$ and $g(x)$ are piecewise functions. For simplicity, we can assume that $f(x)$ has $m$ pieces, and $g(x):=\max_{i=1,2,\ldots, I}\left\{k_i~ x+b_i\right\}$.I have two questions:Is...
View ArticleLet ${\bf A} =(a_{ij})$ be an $m\times n$ matrix such that all $a_{ij}$s are...
Also $\mathbb{E}[a_{ij}]=0$ and Var$(a_{ij})=1$. Let ${\bf x}\in \mathbb{R}^n$. Show that $\mathbb{E}[\|{\bf Ax}\|]=m\|{\bf x}\|^2$ where $\|{\bf x}\|^2={\bf x^Tx}$.
View ArticleIs the Expected Value of $E(Y|Y)=Y$
everyone,I wanted to ask if the expected value of $E(Y|Y)=Y$ ?we know that $E(Y|Y)= \frac{\sum_{y}y p(y,y)}{\sum_{y}p(y,y)}$ but is this equal to Y?
View ArticleExpected value of product of two Ito Integrals
Let $W_1(t)$ and $W_2(t)$ are two standard Brownian motion and $dW_1(t)dW_2(t)=\rho dt$. Calculate $$\mathbb{E}\Big(\int_0^t e^sdW_1(s)\int_0^t e^sdW_2(s)\Big)$$It would be much appreciated if anyone...
View ArticleExpected number of die rolls to get 6 given that all rolls are even.
A fair 6-sided die is rolled repeatedly in till a 6 is obtained. Find the expected number of rolls conditioned on the event that none of the rolls yielded an odd numberI had tried to figure out what...
View ArticleHow to find examples of $L^p$ converging random variables where a specified...
Background: I am preparing for a probability theory exam, and am struggling with a particular type of problem. The questions involve showing that if $X_n \xrightarrow{L^p} X$, then it does not...
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