Expected value of number of specific cards in starting hands in a card game
I don't understand what I found when calculating the expected value of a card game.A deck contains 40 cards. 8 of them are red cards and 32 of them are blue cards. At the start of the game, 5 cards are...
View ArticleExpected value of copula density function
I have a coupla density function $c(u, v)$ (I can assume that $c(u, v)$ is continuous). I want to prove that there is an upper bound $M$ for the following expected value:$$ \mathbb{E} (|\log c(U_1,...
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Why is my approach incorrect (threshold for Dice roll)?
My initial approach to this problem was to fix a threshold $t$, where if your first roll $\leq t$, guess that 2nd roll $>$ first roll, and vice versa. I tried to find the EV of the game as a...
View ArticleHow can we compute the Expectation of Log of Truncated Gamma over (0, 1]?
How can we compute $\mathbb{E} \left[ \log \text{Gamma}_{(0, 1]} (\alpha, \beta) \right]$?Right now I'm using a Monte Carlo estimator by sampling from the Truncated Gamma distribution over $(0, 1]$...
View ArticleExpected number of runs of heads/tails flipping a coin.
Let's say you have a coin that shows heads with probability $P(Heads)=p$.We can show pretty easily (using indicator variables, counting the number of changes) that the expected number of runs is $1 +...
View ArticleExpectation of the product between two dependent Bernoulli random variables
I have two dependent Bernoulli random variables $X$ and $Y$, and I know that:$$\begin{align}P(X=0)&=P(X=1)=1/2 \\P(Y=1)&=11/24 \\P(Y=0|X=0)&=1/3 \\P(Y=1|X=1)&=1/4 \\\end{align}$$Is it...
View ArticleUnderstanding expectation in a single-period market model
I'm auditing a course on stochastic finance. In there a problem was presented, the solution to which wuld be presented on the tutorial I have no access to as an auditor.The problem goes a as...
View Articleconvergence a.s. and expectation of absolute value
I have this kind of question:$\{X\}^n_{i = 1}$ be iid r.v. and $\frac{1}{n}\sum\limits_{i = 1}^{n}X_i \xrightarrow{n \to \infty} 1$ almost surely. Prove that $\mathbb{E} |X_1| < \infty$ and...
View ArticleHow does the order of $\mathbb{E}(X_n)$ impacts the distribution of $X_n$?
Suppose $X_n$ is a sequence of positive random variables with $\mathbb{E} X_n=f(n) = C n^\delta$ with $C>0$ and $Var(X_n)=M=\text{const}$.What can we say...
View ArticleBinomial distribution - Probability of being bigger than the expected value
Suppose we have a random variable X with binomial distribution B(n, p). I'm interested in the probability P(X ≥ E[X]). In particular, is there some inequality for how this probability changes as n and...
View ArticleConsider a continuous random variable 𝑋 with probability density function...?...
2nd Year Engineering Math Question
View ArticleFind $E(S_N/S_0)$ and $E(\frac{S_N^2}{S_0^2})$ for an asset price model
We have the model$$\frac{S_n - S_{n-1}}{S_{n-1}} = udt + \sigma dt^a Y_n,$$with $T=Ndt$, $a =[0,1]$, $\sigma >0$, $P(Y_n=1) = P(Y_n=-1) = \frac{1}{2}.$I have found that$$\frac{S_N}{S_0} =...
View ArticleTailsum Formula and Indicator Functions
In my probability theory class we proved that $$\mathbb{E}[x]=\int_0^\infty \mathbb{P}(X>t) dt,$$ where $X\geq0$ is a non-negative random variable and $\mathbb{E}[X]:= \int_\Omega X(\omega)...
View Articlelimit condition implies finite n th moment for random variable
Problem. Given a random variable $X$, there exists a constant $a>1$ which satisfies $$\lim_{n\to\infty} \frac{\mathbb{P}(|X|>an)}{\mathbb{P}(|X|>n)} = 0,$$prove that $\mathbb{E}(X^p)$ is...
View ArticleHelp with understanding the calculation of an expected sum
A box with $N$ balls numbered from $1$ to $N$. We take $n$ balls out with no returning. I need help with understanding the calculation of their expected sum.So let's say that $S_n$ is the sum of the...
View ArticleExpectation of a sample space given the expectation of each block of a...
Suppose $A$ is a sample space, which is partitioned into sample spaces $A_1,A_2,...A_n$. Suppose I knew the expected values of a random variable on each $A_i$. is it true that the expected value of...
View ArticleOnce $X\sim \text{gamma}(\alpha = 12, \beta = 2)$ is observed, $Y$ is...
Random variable $X\sim \text{gamma}(\alpha = 12, \beta = 2)$. Once $X=x\gt 0$ is observed, $Y$ is randomly chosen from $(0,x)$. Evaluate $E(Y)$.Well I know $E(Y)=E_X[E(Y|X)]$ so I'll proceed by...
View ArticleConditions such that $\text{Cov}(X, ZY)= 0$ for $X\sim Y \sim U(-1,1)$ and...
Say I have three random variables $X\sim Y \sim U(-1,1)$, and a third one $P(Z=1) = P(Z=-1) =1/2$. What condition on these three variables must I have so that I can say $\text{Cov}(X, ZY)= 0$? I...
View ArticleBasketball Betting Pool - Expected Number of Points...
There are 64 teams who play single elimination tournament, hence 6 rounds, and you have to predict all the winners in all 63 games. Your score is then computed as follows: 32 points for correctly...
View ArticleProve that...
Let $X, Y$ be two independent random variables, both uniformly distributed on $[0, 1]$ and let $g:\mathbb{R}\rightarrow \mathbb{R} $ be continuous function. Does following inequality always...
View ArticleExpected Number of Points in Two-Player Dice Game
I came across this problem online and it has been in the back of my mind ever since. I'm currently taking a probability course so please forgive my limited knowledge if I'm missing a key...
View ArticleExpected number of consecutive heads in 10 coin tosses
I am having trouble formulating the exact recursive relation for this problem. The problem statement is A coin is tossed 10 times and the output written as a string. What is the expected number of HH?...
View ArticleExpectation of numbers of circles in a banquet. [duplicate]
At a banquet, there are $n$ people who shake hands according to the following process: In each round, two idle hands are randomly selected and shaken (these two hands are no longer idle). After $n$...
View ArticleWhat is the expected number of rolls to get every number at least twice?
How many rolls will it take on average to see every value of a sixsided fair dice at least twice?For seeing every value at least once, this question seem much more straight forward as you can think of...
View ArticleExpected number of rolls until all dice are removed (followup)
Suppose we roll n dice, remove all the dice that come up 1, and roll the rest again. If we repeat this process, eventually all the dice will be eliminated. How many rolls, on average, will we make?I...
View ArticleDerivation of Expected Number of Occurrences in a stochastic Intensity...
I have a question concerning the expected number of occurrences in a random intensity Poisson process during a specific interval.Let $N_t(h)$ be a random variable counting the number of events that...
View ArticleBasketball Expectation Value question
ContextHere is the problem I was working on: A player is shooting free throws. They make the first, miss the second, and from then onwards, the probability that they make a shot is the fraction of...
View ArticleExponential Moment of Uniform Random Vector
Let $Z = (z_1,\dots, z_d)$ with $z_i$ i.i.d. uniform random variables on the interval $[a,b]$. I am interested in computing$$\mathbb{E} [\exp (\|Z\|^2)] = \mathbb{E} [\exp (z_1^2 + \dots z_d^2)].$$My...
View ArticleAny example for $E[(X+Y)^2Z^2]
Let $X,Y,Z$ be three random variables. Is there any example such that$E[(X+Y)^2Z^2]<E[X^2Z^2]$, $E[(X+Y)^2]\ge E[X^2]$, $E[XZ]\ne 0$, and $E[XY]=E[YZ]=E[X]=E[Y]=E[Z]=0$?
View ArticleWhat allows me to write a nested expectation as a sum of expectations?
Assume I have a random process $x_1,x_2,\ldots,x_N$ defined on a probability space $(\Omega,G,P)$.Assume I have the expectation$$E_{x_1}\large[x_1+E_{x_2\vert x_1}\large[x_2+E_{x_3\vert...
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