Why does this equation hold? This is a question about ddpm and nscn. [closed]
Can someone help me why this equation holds:$\nabla _{x_{t}}logq(x_{t})=E_{q(x_{0})}[\nabla _{x_{t}}q(x_{t}|x_{0})]$?This equation is about the connection between Denoising diffusion probabilistic...
View ArticleTruncated distribution and the property of its expected value
I have a complicated random variable $Y$ where the support is on $\mathbb{R}$ (real value). Now, we construct a truncated random variable $\tilde{Y}$ such that$\tilde{Y}=Y$ if $-C_0\leq Y\leq C_1$,...
View Articleexpected value from some points in continuous homogeneous spatial Poisson...
Let $n$ point are distributed as per a homogeneous spatial Poisson process of rate $λ$ in a square of side $2a$, and assume that $4$ fixed points are located at $(\frac{a}{2},\frac{a}{2})$,...
View ArticleProbability that a specific card appears before turning up exactly two aces...
I'm just adopting this new idea of "symmetry" in probability. To solve the question in the title. My reasoning is: we have five places for a specific card(out of $48$ different cards from $4$ aces)...
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Calculation of Standard Deviation
The stochastic variable $X$ has the distribution function:$$F_X(x) = \begin{cases} 0 & \text{if } x < 2, \\ 0.5 & \text{if } 2 \leq x < 4, \\ 0.8 & \text{if } 4 \leq x < 8, \\ 1...
View ArticleExpected number of real roots of a polynomial with independent, uniformly...
What is the expected number of real roots of a degree 4 polynomial whose coefficients are independently, randomly generated with uniform distributions on [0,1]? I did a Monte Carlo simulation which...
View Article$\mathbb{E}[Y \mid X_1] = 1 + (1 - X_1)\mathbb{E}[Y]$ [closed]
Let $X_1,X_2...$ be iid and assume:$P(X_1 = 1) = 1 - P(X_1 = 0) = p \in (0, 1)$Let Y be defined as:$Y = \inf\{n \in \mathbb{N} : X_n = 1\}$So that $Y=n$, where $n \in \mathbb{N}$ if and only if...
View ArticlePlayer in casino
The player plays the following game in a casino. He bets c (some amount in dollars), says a number from 1 to 6, after what croupier rolls three dice. If the number player has said hasn't shown up on...
View ArticleThe sum of decaying log-normal exponentials
For the following expression:$$S=\sum_{i=1}^N \frac{1}{N} e^{-x_it},$$where $N$ is reasonably large (at least $N>10^2$, but particularly interesting are the cases where $N>10^5$), $t>0$, and...
View ArticleWhy is average of a random subset the same of the initial set?
This feels like a really simple question, plus it looks very intuitive, but I can't find a formal proof or anything that proves it.I have an initial set $X$ with $n$ values. I extract $Y$, a random...
View ArticleExpectation of sample variance
Let $s^2$ be sample variance, $\sigma^2$ be population variance$E[\frac{(n-1)s^2}{\sigma^2}] = E[\chi^2_{n-1}] = (n-1) \implies \frac{(n-1)E[s^2]}{\sigma^2} = n-1 \implies E[s^2]=\sigma^2$But if i do...
View ArticleTriangle inequality raised to the power of $n$
Let $r\in [1,\infty)$ and $X$ be a random variable such that $E[X^r]<\infty$. Let $\{a_n\}_n$ be a sequence of positive real numbers such that $\lim_n a_n = 0$. My question is whether or not we have...
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Normal distribution and conditional expectation
Why is $E(X|X>a) \left( 1- \Phi\left( \frac{ln(a)-\mu}{\sigma} \right) \right)=E(X) \left( 1-\Phi\left( \frac{ln(a)-\mu}{\sigma} -\sigma \right)\right)$?Maybe it's easy but I do not see it.
View ArticleExpected 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...
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Sarah the squirrel problem (Linearity of Expectations)
From https://brilliant.org/wiki/linearity-of-expectation/I have two questions:First questionThe "obvious" solution is by using linearity of expectations. My idea was to write the states as $E(n -...
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...
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On average, how many random points can be chosen on a disk, until they can no...
On a disk, uniformly random points are chosen, one by one, until they can no longer all be enclosed by a triangle that lies with the disk. Let $X=$ number of points chosen before the final point.In the...
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 +...
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