Expected payoff of two dice scenario
I've been given the following question:You have two fair $10$-sided dice that are colored green and yellow with values $1$ to $10$ on each side. You roll both dice simultaneously. If they show the same...
View ArticleExpected number of edges required for a graph to have a triangle.
I am considering graphs on $n$ with edges added iid randomly with probability $p$.I have come across this post for the expected number of edges for a graph to have a triangle. In the question, they say...
View ArticleHow can we justify why $\mathbb{E}[X] = 1 + \frac 1 2 \mathbb{E}[X]$ here?
I was wondering what the formal justification is behind the following reasoning:Lemma On average a fair coin needs to be tossed two times before a "heads" is seen.Proof. Let $E$ be the expected number...
View ArticleAsymptotic distribution of repetitions in binary sequence
Let $N$ be a fixed positive integer.Let $b_1,\dots,b_M$ be a sequence of i.i.d. Bernoullay random variables with $p=\frac12$. So $P(b_i=0)=\frac12=P(b_i=1)$ for all $i$. Now let $X$ be the amount of...
View ArticleFraction of times Alice redraw from N(0,1)
Alice and Bob are playing a game. Each of them draw a random number (one by one starting with Alice) from a standard normal distribution. They are not allowed to see the opponent's number, but are...
View ArticleEfficient way to find the expected value of the exponent of two 6 sided die
Given you roll two fair six sided die x and y with faces from 1 to 6, is there any trick to calculate the expected value of x^y?I believe you could calculate the expected value of each of 1^y, 2^y ......
View ArticleExpectation Question of Dice roll [closed]
Suppose we play a game. I roll a die up to three times. Each time I roll, you can either take the number showing as dollars, or roll again. What is your expected winnings?
View ArticleExpected value of the maximum of the absolute value of two random variables
Given a stochastic process $\{X(t), t > 0\}$, defined by $X(t) = At + B$ with two random variables $A$ and $B$ which are independent normally distributed random variables with mean $0$ and variance...
View ArticleExpected Breakup Length [duplicate]
The question - A stick is broken into 3 pieces, by randomly choosing two points along its unit length, and cutting it. What is the expected length of the middle part?So the answer using symmetry is...
View ArticleHow do we prove for "most" $f$, the expected value of bounded sequences of...
Let $n\in\mathbb{N}$ and suppose function $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$, where $A$ and $f$ are Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, where...
View ArticleExpected Number of Flips and Probability in a Coin Toss Experiment
I'm currently studying probability and I've come across a problem that I'm finding quite challenging. I would appreciate any help or guidance.Problem Statement:A fair coin is continually flipped until...
View ArticleFinite Expectation Leads to Finite Entropy [closed]
Let $X$ be a random variable supported by the naturals, with a finite expectation. Show that $H(x)<\infty$. Also, give an example of a random variable supported by the naturals that has...
View ArticleOptimal bet using the Kelly criterion if the odds and probability update.
I'm wondering about how to change the bets given that our information about the probability and odds may change throughout the event.For example, let's say we have \$100 to bet, and there is an event...
View ArticleUsing the Moment Generating Function to find the Expectation of a non linear...
Let $X$ be a r.v which has the following moment generating function $$M_{x}(t) = (1-2t)^{-2}$$If $Y = (100)(1.1)^{X}$ then calculate $E[Y]$I know that $X \sim \chi^{2}_{(4)}$ but I am supposed to...
View ArticlePascal gets drunk and builds his triangle. Half the time he writes $0$...
One day Pascal has a little too much to drink, then sits down to build his famous triangle. He writes $1$'s going down the sides. Then he starts filling in the triangle. Each number is supposed to be...
View ArticleWhat is the expectation of the Dirac delta function of a random variable
I have a Dirac delta function as follows:$\delta_{\epsilon = y}$ where $\epsilon \sim N(0, \sigma^2)$, and $y \in \mathbb{R}$. I want to take the expectation of this function, but it appears that it...
View ArticleExpected depth of random tree which randomly select the parent node for every...
Consider using the following method to generate a tree:Set node $1$ as the root;For node $i$ ($i \in [2,n]$), randomly select the parent node in $[1,i)$.(The expression of the interval here is not...
View ArticleFind $E(|X-Y|)$ Given $E(X)$ and $E(Y)$ [closed]
I have a question involving two random variables involving X and Y, where the expected value of each is known.How can I find E(|X-Y|)? Is the information given enough, or should I need any other...
View ArticleExpected Trials for Dual-Condition Set Partitioning
If we have a set of 200 elements that are repeatedly randomized in order, then split into two groups: one group contains 80% of the elements, and the other group contains the remaining 20%. This...
View ArticleExpected Value of Posterior distribution where the function has cases.
Assume I have a prior over two random variables $\theta_A \sim U[0,\hat{\theta_A}], \theta_B \sim U[0,\hat{\theta_B}]$. Let's say I observe $x = 1$ if $\theta_B b + \theta_A a > z$ and $=0$...
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