Expected sample maximum for discrete distribution
Given a non-uniform discrete distribution:$$P(x) = \begin{cases} \frac{1}{30}, & \text{if } x = 4 \\ \frac{4}{30}, & \text{if } x = 3 \\\frac{15}{30}, & \text{if } x = 2 \\\frac{9}{30},...
View ArticleExpected time in half-line for random walk
For a one-dimensional random walk (starting at $0$) for which we move $1$ unit to the right with probability $p$ and $1$ unit to the left with probability $q=1-p>p$, what is the expected time spent...
View ArticleExpectation and the d'Alembertian in $M^2$ [closed]
I am currently reading "Does Locality Fail at Intermediate Length-Scales?" by Rafael Sorkin. My question concerns equation $(3)$ in his paper. How does the expectation of the linear operator $B_{xy}$...
View ArticleThe expectation of absolute value of the sum of n i.i.d. random variables...
Let $\varphi_i$ be a Gaussian random variable such that$$\varphi_i \sim N(0,\sigma^2), \quad i = 1,2,\ldots,n.$$What's the expectation:$$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |\right)...
View ArticleCalculate the variance and expectection of $\hat{y}$ in a linear regression...
I have the following linear regression model$$y = \beta_0 + \beta_1 \cdot 40$$where $\beta_0 = 11.1317$, $\beta_1 = 1.01$, and $40$ is simply the value of the predictor variable (I guess). As you can...
View ArticleMinimum expected number of 5-die rolls to simulate a fair 6-die
Context: If we only have a fair 5-sided die, we can emulate a fair 6-side die roll by rolling D5 twice. Out of 25 outcomes, we can map 24. If we get the 25th outcome, eg. (5,5), we just repeat the...
View ArticleDefinition and derivation of conditional expectation/probability
I read quite a few books introducing the notion of conditional probabilities/expectation by putting a formula out there coming from what they call "intuition".Can someone provide me a good measure...
View ArticleHow to derive $\mathbb{E}[X \mid Y]$ when $Y(\omega) = \omega(1-\omega)$ on...
I am working on the following past paper problem:Let $\Omega = [0, 1]$ with $\mathbb{P}$ the Lebesgue measure on $[0, 1]$. Suppose that $X$ and $Y$ are random variables on $(\Omega, \mathbb{P})$, and...
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Minimal conditions required to randomly form a sphere
Question Let $X_1,X_2,\cdots $ be i.i.d. random variables with values in $S^2:=\{x\in \mathbb{R}^3:|x|= 1\}$. Then with $C_k:=\{x\in \mathbb{R}^3:|X_k\times x|\leq 1\}$ being a cylinder in the...
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Is this just a coincidence? Expectation of product of areas in unit circle...
Draw line segments from the centre of a unit circle to two uniformly random points on the circle, forming two regions of area $A_1$ and $A_2$.It is easy to show that the expectation of the product of...
View ArticleWhy is the expected value of this random nested radical equal to $4/\pi$?
To celebrate Pi Day, I took a look at some classic identities for $\pi$ again, particularly the nested roots associated with Viète’s formula. Such as this one:$$\pi=\lim_{k \to\infty} 2^k...
View ArticleUsing the law of iterated expectations with multiple variables [closed]
I'm given that:$E[\epsilon_i | dist_i, range_i] = 0$How can I apply the law of iterated expectations to find that $E[\epsilon_i|dist_i] =0$is also true?
View ArticleAnalytical methods to derive expected points from $\texttt{xG}$ differential...
In football, Expected Goals ($\texttt{xG}$) quantifies the likelihood of scoring from shots, where a team’s $\texttt{xG}= k$ is the sum of probabilities $\left\{ p_{1}, \ldots, p_{N} \right\}$ of $N$...
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Expected number of collinear points in non-repeating random walk (from XKCD)
Today's XKCD-comic (see below) featured a couple of joke open math problem, and one of them caught my interest (being the only one that's an actual mathematical question!):If I walk randomly on a grid,...
View ArticleExpected Number of Coin Tosses to Get Five Consecutive Heads
A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
View ArticleProving formula of expected Value
I want to proof the formulas for the expected Value of discrete and absolutely continuos Random Variables $\mathbb{E}[X]=\sum_{i=1}^\infty x_ip_i$ and $\mathbb{E}[X]=\int_\mathbb{R}xf_X(x)dx$by using...
View ArticleExpected number of trials for success vs. number of trials for expected 1...
Consider a sequence of trials $1, 2, \ldots$. We assign a probability of success $p_i$ to trial $i$. (For a less constrained version, we might suppose that $p_i$ is the probability of success given...
View ArticleExpectation value of exponential of a function - first two moments of...
I have a function $f(t)$ and know that $\langle f(t)\rangle=0$ and $\langle f(t)f(t')\rangle=C(t-t')$, where $\langle~\rangle$ denotes the expectation.Now I want to...
View Article100 prisoners and a lightbulb
100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb...
View ArticleI roll a die repeatedly. Each roll I can choose to keep the die's value in \$...
I roll a die repeatedly. Each roll I can choose to keep the die's value in \$ and stop rolling, or pay \$$1$ to roll again. What's the best strategy?It seems obvious to me that at each stage, if your...
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