Expectation 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 ArticleHelp me to determine E(N|D=d), I do this know how to develop more [closed]
I have a random variable N, that follows a Poisson distribution, with parameter $\lambda$. I have another variable D that follows a Poisson distribution, with parameter $p \lambda$. The probability of...
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...
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