Questions tagged [expected-value]
Questions about the expected value of a random variable.
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Expectation of squared normal cdf
Let's assume a non-linear, strictly decreasing function of a random variable $Y\sim N(0,1)$ of the form :
$$p(Y) = \Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho} Y}{\sqrt{1-\rho}}\right)$$
where $\Phi(\cdot)...
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Finding E[Z²Φ(Z)] using law of the unconscious statistician
Question:
Random Variable Z follows a standard normal distribution with c.d.f. Φ.
Find E[Z²Φ(Z)] using the law of the unconscious statistician.
I am thinking of using some substitution to work it out, ...
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Is an expectation over two variables equivalent to the expectation over the first of the expectation over the second?
My question is in regards to this surrogate advantage function used in the Trust Region Policy Optimization (TRPO) reinforcement learning algorithm.
$${L}(\theta_k, \theta) = \mathop{\mathbb{E}}_{s,a \...
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Linear Regression Seber - Why is the matrix A written in this way?
Example 1.9 of Seber’s book states the following:
The result stating that if $\rho = 0$ it holds that $Q = \sum _i (X_i - \overline X)^2$ has expected value $\sigma ^2 (n-1)$ follows from $\...
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Expectation of a random variable related to expectation of its indicator function
Let $X^* $ be a random variable and let $X = 1[X^*>0]$. Where $1[.]$ is the indicator function.
I am interested to express the expected value of $X^*$ in terms of the expected value of $X$. In ...
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Relating expectation of a difference of random variables with expectation of their indicator function
Let $X^*$, $Y^*$ as two random variables; respectively with pdfs $P_{X^*}$ and $P_{Y^*}$
Let $X = 1[X^*>0]$ and $Y = 1[Y^*>0]$. Where $1[]$ is the indicator function.
I am interested in finding ...
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Dominance of exponential moment due to Stochastic dominance
Suppose $X$ and $Y$ are positive random variables such that $P(X\geq x)\leq P(Y\geq x)$ and let $\lambda >0$
Then can we say $$E(e^{\lambda X})\leq \int_{0}^{\infty}e^{\lambda x}P(Y\geq x)\,dx $$
...
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Density of normal c.d.f. of a normal random variable
Let $a$ be a constant, X is a standard normal and $\Phi$ is the c.d.f. of standard normal.
what is the density of $Y$ where $$Y = \Phi(aX)$$
what is the density of $$Y=\Phi(X + a)$$
The second case ...
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Expected value of a continuous random variable must be strictly bounded
Given a probability space $(\Omega, \mathcal{F}, P)$ and a continuous random variable $X: \Omega \to I$ where $I$ is an interval of $\mathbb{R}$. I'm trying to show that the expected value $$E[X] = \...
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Finding maximum value using expected profit with Binomially distributed demand
Let $D$ be Binomially distributed with mean $25$ and variance $20$. This means that $D\sim\text{Bin}(125,\frac{1}{5})$. I need to determine the optimal order quantity $Q$ which maximizes profit. For ...
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Do the projection errors resulting from linear projections onto each other satisfy any equality restriction?
Suppose I have two random variables $X_1$ and $X_2$ that are not independent of each other. Consider the linear projections of $X_1X_2$ and $X_1$ onto each other: $L(X_1X_2|X_1)$ and $L(X_1|X_1X_2)$, ...
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Recursive Uniform Distribution Expectation Question
Suppose we draw some k ~ Unif(0, 1). Then, we will draw some $u_1$ ~ Unif(0, 1). If $u_1 < k,$ we stop. Else, we will draw $u_2$ ~ Unif(0, $u_1$). We will continue drawing until $u_n < k.$ What ...
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Expectation of sum of indicator function on uniform random variable
Let C be an estimator defined as
$$C = \frac{1}{n}\sum_{k=0}^{n}1_{U_k>a}$$
where $a$ is a real number in $(0,1)$ , $1$ is an indicator function and $U_k$ are uniform random variable in $(0,1)$.
I ...
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Same expectation on all sets in a $\pi$-system means equal almost surely
I think I have a good intuition on this problem but would appreciate a second look as to whether I'm missing any details, particularly the algebraic details.
Problem: We have $L_1(\Omega, F, \mathbb P)...
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Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?
Let's take a random walk on Pascal's triangle, starting at the top. Each number is in a regular hexagon. At each step, we can move to any adjacent hexagon with equal probability, but we cannot revisit ...
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Expected number of strikes to kill a $3$-headed dragon
You want to slay a dragon with $3$ heads. There is $0.7$ chance of destroying a head and $0.3$ chance of missing. If you miss, a new head will grow. $X$ is a random variable for the number of rounds ...
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Expected winning score in a round robin tournament
Suppose you have a tournament consisting of 100 players in which each player plays each other player exactly once. In any given match, the chance of either playing winning is equally likely, and there ...
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Finding expected value with replacement [closed]
I have balls numbered 1 to 100 in a bag. I take one out, record it, and put it back into the bag. I repeat this 1000 times. My questions are:
What is the expected value E[X] of never picking a number?...
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Covariance of the product of two random variables with another random variable
Let $X\sim \mathrm{Bin}(p_x,1)$ , $Y\sim \mathrm{Bin}(p_x,1)$ and $V\sim \mathrm{Bin}(p_v,1)$ be three binomial random variables, that are NOT independent from each other. Note that $X$ and $Y$ are ...
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Show that $L^p$ convergence is unique almost surely
It'd be of great help if someone could double-check if my proof is correct/rigorous.
Problem: If $X_n\to X$ and $X_n\to Y$ in $L^p$, then $X=Y$ almost surely
My proof: We have that $\lVert X_n -X \...
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Expectation of matrix-matrix-matrix product
When calculating the vector-matrix-vector product
$$\mathbb{E}\left[x_1^TWx_2\right],$$
where $x_1$ and $x_2$ are $n \times 1$ vectors of random variables and $W$ is a $n \times n$ constant matrix, ...
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Expected Value of Drawing Cash from Bins.
A bin consists of 4 \$10 bills, 3 \$20 bills and 1 \$100 bill. You select 7 bills out of this bin uniformly at random without replacement. Find the expected value of the sum of all the bills you ...
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When does $k$-th central moment relate to $k$-th power of expectation?
Given a bounded random variable $X$. When exactly is the $k$-th central moment $ E (X-\mu)^k $ upper bounded by the $k$-th power of the expectation $ \mu^k$? This seems to be the case for bounded, ...
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Limit of expectation of reciprocal mean of uniforms
Suppose we have a sequence of iid unif$(0,1)$ random variables. I want to know whether or not the sequence of $\mathbb E[\frac1{\bar X_n}]$ converge to 2 (since $\bar X_n$ strongly converges to $\...
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Another formula for Expectation
I came across the following question:
If $ X $ is a continuous random variable $ (-\infty<X<\infty) $ having distribution function $ F(x) $, show that $$ E(X)=\int_{0}^{\infty}[1-F(x)-F(-x)]\,\...
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Upper bounds for E(f(x))
Given a discrete random variable X, with finite mean $\mu$ and variance $\sigma^2$ and a convex function f(x), what is the tighter upper bound one can give for E(f(x))? I am looking for something in ...
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Find variance and expectation with given probabilities of variable values
I've got random variable $x$ and probabilities: $P(x=0)=1/4, P(x\in[a,b])=\cfrac{b-a}{2}, P(x=3)=1/4$ where $1\leq a < b \leq 2$.
What I did:
First I tried to calculate the mathematical expectation ...
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prove $E\frac{X^k}{\omega X^{2r}+\sigma}<\infty$ for $\omega,\sigma>0, k\in\{0,1,...,2r\}$, $r\in \mathbb{N}$
I have found a statement here equation 23 without explanation that
$E\frac{X^k}{(\omega X^2+\sigma)^r}<\infty$
for $\omega,\sigma>0, k\in\{0,1,...,2r\}$, $r\in
\mathbb{N}$, where we don't know ...
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Sub-exponential Norm of Normal Distribution
Let $X \sim \mathcal{N}(0, \sigma^2)$. Define the sub-exponential norm as follows for a random variable $Z$:
$$ \|Z\|_{\psi_1} := \inf \left\{k>0 \vert \mathbb{E}\left[\exp \frac{|Z|}{k}\right] \...
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Maximum of Sub-Exponential and its Tail Probability
Consider zero-mean sub-exponential random variables $\{X_1,...,X_n\}$ (not necessarily independent) with parameters $(\nu, \alpha)$. That is,
$$\mathbb E[\exp\{\lambda X\}] \le \exp\{\nu^2\lambda^2/2\...
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Find conditions on $f,g$ $E[ \langle g(X), f(X) \rangle ] \ge \langle E[ g(X)], E[f(X)] \rangle $ (i.e., FKG inequality for multivariate functions))
The classical FKG inequality states that for two non-decreasing function $f,g: R \to R$ we have that
\begin{align}
E[ f(X) g(X) ] \ge E[f(X) ] E[g(X)]
\end{align}
I am interested in understanding ...
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Marginalization of the conditional expectation
Let $Y, Z, X_1, X_2, T$ be random variables. Let $\mathbb{E}(Z\mid X_1, X_2) = \mathbb{E}[Y\mid T=t, X_1, X_2]$. I need a property
$$\mathbb{E}(Z\mid X_1) = \mathbb{E}[Y\mid T=t, X_1].$$
Does it hold ...
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Lower-bound for $E(\log((x+1)!))$? [closed]
I have this random discrete variable $X$ such that $E(X)=\mu$. Is there a closed form for the value of $E(\log((x+1)!))$? If not, is there a lower bound for it, for example using Jensen inequality? I ...
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$x$ is distributed as Poisson with parameter $\lambda$, prove $\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0$
Let $x\sim\text{Pois}(\lambda)$, prove that for any $\lambda>0$,
$$f(\lambda)=\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0.$$
In other words,
$$\sum_{x=0}^{\infty}\log(x+0.5)\frac{\lambda^x}{...
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What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
Intro__________________________
I was studying in Youtube this interesting MIT course of math ...
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Find this expected value [closed]
Let $\xi_1,\xi_2,\cdots, \xi_n, \cdots$ be a sequence of identically distributed continuously randomly variables and $\nu = \text{min}\{k: \xi_{k-1} > \xi_k\}$. Find $\mathbb{E}(\nu)$
I'm ...
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Expectation of the scalar product between matrices
Let $M=W_1 + d W_2$ and $M^*=W_1^* + d W_2^*$ with $W_1, W_2, W_1^*, W_2^* \in \mathbb{R}^{rxr}$. We define $\mu = \sum_{I=1}^{n}x_i$ with the $x_i$ are drawn from $\mathcal{N}(0,I)$.
It is written in ...
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Expected value - No consecutive heads sequence
Problem: Suppose that you flip a coin $n$ times and it turns out that
no consecutive heads appear in the sequence. Let $E(n)$ be the
expected number of heads that appear given this information. Find $
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How should I play this dice-rolling game with modifications?
I have an $100$-sided die. Every round, I roll the die, and can either choose to receive the amount shown by the die, or pay $1 to reroll the die and continue.
If I want to find the expected value of ...
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Conjecture: If $x_k$ are random in $(0,\pi/2)$ then expectation of $\frac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ is $(\pi/2)^{2n-6}$ for $n>2$.
Let $E(n)=\text{expectation of }\dfrac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$ where $x_k$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.
Is the following ...
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Probability and average height of two lines intersecting above the x-axis
For context, I know at least basic calculus and what random variables are, but not much about doing calculations with random variables. I came up with a problem for myself where there are two points ...
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Card game Puzzle
You have a special deck with 11 cards, with 10 red cards and 1 joker. starting with $1 we play a game:
shuffle the deck
we can choose to draw the top card. if it's red, we double our money, but if it'...
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Question about Expectation
Is my thinking correct?
Let's suppose there are two independent random complex vectors, $a,b\in\mathbb{C}^N$. Additionally, both vectors have a zero mean, and their covariance matrices are defined (e....
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$\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? Seriously?
In the diagram, $\alpha$ and $\beta$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$.
What is $\mathbb{E}(h)$?
Superimposing a cartesian coordinate system, the ...
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Product of 2 dice game
You are playing a game where you a paid the product of 2 dice
a) How much would you pay to have the opportunity to reroll any 1 of the dice?
b) How much would you pay to roll 3 dice and discard the ...
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Determining bet sizes given odds [duplicate]
Crossposted on Quant SE
Recently, I was asked the following question in an interview with a prop trading firm.
You are given the opportunity to make money by betting a total of 100 bucks on the ...
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Does $E^2(X)/E(Y) \le E(X^2/Y)$ hold
Let $f(x),g(x)$ be $\mu$ measureable functions with $g(x) > 0$ almost surely.
Does $$\frac{\left(\int f(x)\mu(x)dx\right)^2}{\int g(x)\mu(x)dx} \leq \int \frac{(f(x))^2}{g(x)}\mu(x)dx$$ hold?
I ...
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Probability of guessing correct number between 1-100
You need to guess a number between 1-100 and after each guess I'm going to tell you if you guessed too high or too low. If you guess it on the first try you get \$5, on the second try \$4, then \$3, \$...
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Expected value to have no heads
This question was asked in an interview of a Quantitative finance company.
You have $\textit{n}$ fair coins with either heads or tails facing up arranged in a row.
You play a game in rounds described ...
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toy collection question
Every box of cereal contains one toy from a group of five distinct toys, each of which is mutually independent from the others and is equally likely to be within a given box. On average, how many ...
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