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Questions tagged [expected-value]

Questions about the expected value of a random variable.

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Bound on the expectation of the maximum of a sequence, given bounds on the expected value of each element.

I have a sequence of independent random variables $U_1, U_2, \dots,U_N$. Suppose $\mathbb{E}[U_i] \leq 1$ for all $i=1,\dots,N$, and let: $$M_N = \max_{i=1,\dots,N} U_i$$ It is easy to see that $\...
Uomond's user avatar
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2 votes
2 answers
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Optimal strategy for uniform distribution probability game

There are 2 players, Adam and Eve, playing a game. The rules are as follows: $n$ and $d$ are chosen randomly. Adam samples a value $v$, distributed uniformly on $[0,n]$, and can either cash out $v$ or ...
jimsimons's user avatar
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1 answer
24 views

dividing the determinant of a matrix by the expected value of the determinant of the same matrix over a uniform distribution

Let $A, B$ be square $n \times n$ matrices as follows: $$ A = \begin{bmatrix} x_1&x_2&\cdots&x_n\\ x_{n+1}&x_{n+2}&\cdots&x_{2n}\\ \vdots&\vdots&\vdots&\vdots\...
M a m a D's user avatar
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7 votes
3 answers
878 views

This expected value has a minimum!

Problem. Let $X$ be a positive, real random variable whose probability density function is bounded by $1$. Prove that $E[X]\geq \frac 12$. Hi everyone. This problem is essentially saying that the ...
aleph2's user avatar
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-1 votes
1 answer
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Probability - Find the expected number of item throws into N containers until one of the container reaches k items [closed]

There are $N$ empty containers, which are unlabeled and are exactly the same. Each time, with equal probability, a ball is throwed into a random container. Q: What is the expected number of throws $f(...
Froest's user avatar
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1 answer
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What is the uses of Expected Value in this context?

I'm reading The Element of statistical learning: https://hastie.su.domains/ElemStatLearn/ and having question regarding this example on pages 23 and 24: "Suppose we have 1000 training examples $...
alksdhalksjdb's user avatar
1 vote
1 answer
53 views

Conditional expectation - alternative expression

Consider the following set-up. $F:[0,\omega]\rightarrow[0,1]$ where $X$ is a real-valued random variable. The conditional expectation of $X$ given $X<x$ is: $E(X|X<x)=\frac{1}{F(x)} \int_0^s tf(...
Frank Swanton's user avatar
1 vote
1 answer
38 views

Doubts on "An Intensive Introduction to Cryptography" exercise about Shannon's entropy

I was going through the exercises in An Intensive Introduction to Cryptography (see full PDF here), and in particular, I had some doubts on Exercise 0.12 (found on page 42). Here is the relevant ...
chirpyboat73's user avatar
1 vote
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21 views

Upper bound for distribution function for variable with zero expectation. [duplicate]

A problem from final Year 1 probability exam. Is it true for any random variable $Y$ s.t. $E[Y]=0$ and $E[Y^2]<\infty$ that: $P(Y>x)\leq\frac{E[Y^2]}{E[Y^2]+x}$ ? I thought we can rewrite it ...
innerproduct's user avatar
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Expected Number of Letters Typed Until MOO is Typed When Letters Are Typed Randomly

I'm failing to see the mistake in my reasoning for this problem. Here is the problem: Problem A man can only type two letters: M and O. He types M with probability $.4$ and types O with probability $....
Goku241's user avatar
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1 answer
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Question on the expected number of same color balls left in a urn

I'm working on a problem where I am given an urn with $a$ white balls and $b$ black balls. One ball at a time is selected randomly until there is only balls of the same color. I am asked to find the ...
Kham Bodrogi's user avatar
3 votes
0 answers
37 views

$L_2$ convergence of bivariate function

I have the following problem: Let $X,Y$ be random variables with distributions $P_X,P_Y$ and $f_0$ be a map from the support of X,Y to the reals. I define a new function $\chi_0(y) = E_X[f_0(X,y)]$. ...
xcesc's user avatar
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0 answers
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Expected number of closed shapes in an $n\times n$ grid with some lines missing

I came up with a math puzzle that I can't figure out how to solve. I feel like it has enough "math" to make it more appropriate to post here than to the Puzzling Stack Exchange. Here it is (...
Dylan Levine's user avatar
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Which distributions have nice closed form expressions for expected value of exponential?

which distributions have nice closed form expressions for $e^{-kx}$ and $xe^{-kx}$, where $k$ is some known constant? Ideally the support of the distribution should be positive, so for example the ...
Jacques La Fontaine's user avatar
3 votes
0 answers
55 views

Expected number of edges to draw in a bipartite graph until you get a crossing

I was asked by a friend to calculate the number of edge crossings in a $m \times n$ complete bipartite graph: Now play a game where you randomly select an edge with equal probability each turn: what ...
lnx's user avatar
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1 answer
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Linearity of expected value and what it implies

In some problem I was doing, it involves a trial where given some value $x$, $50\%$ of the time it will go to $0$ at the next step, and $50\%$ of the time it will go to $2x$. Notation-wise, is it fine ...
Aiden Chow's user avatar
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9 votes
1 answer
165 views

Upper Bound on $\mathbb E[\sqrt{X}]$ with $X\sim\operatorname{Bin}(n,\frac{1}{n})$ that is not Jensen

Let $X_n\sim\operatorname{Bin}(n,\frac{1}{n})$ a random variable with binomial distribution. We then have $\mathbb E[X_n]=1$. My goal is to find a cosntant $0< c< 1$, s.t. \begin{equation} \...
Joseph Expo's user avatar
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Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function

In a proof of Jensen's Inequality that I am reading, the following is used: If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$. Why can we deduce ...
guest1's user avatar
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1 answer
60 views

expected value of high-low guessing game

Assume a number between 1-100 (inclusive) is chosen randomly. You then attempt to guess the number. On each guess, if you didn't get the exact number, you're told whether the guess is higher or lower ...
james's user avatar
  • 15
2 votes
2 answers
66 views

Expectation of a non negative integer value random variable

the expectation for a discrete type random variable that takes non negative integer value is given by: $$ E[X] = \sum_{k=0}^\infty kP(X=k) = \sum_{k=0}^\infty P(X>k) $$ The derivation involves the ...
Kumar Yashasvi's user avatar
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1 answer
39 views

Limit of expectation of ratio of sums of uniformly distributed variables

I saw this problem in past year exam of probability theory. Let $U_1, \ldots, U_n, \ldots $ be independent standard uniformly distributed variables: $U_i \sim Uniform(0, 1)$ and $a>b>0$. ...
innerproduct's user avatar
0 votes
0 answers
45 views

Stopping time for uniform law

Let $X_1, X_2, \dots$ be IID Unif$(0,1)$ random variables and let $N=\min \{n : S_n=X_1 + \dots + X_n > \ln(2) \}$. Find the expectation of $N$. I've tried three approaches. First I showed that $...
Kilkik's user avatar
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9 votes
4 answers
907 views

Can the differential be unitless while the variable have an unit in integration?

Apologies for terminology inconsistencies, as I'm reading a Chinese statistics and probabilities textbook while looking up intrinsics on an English encyclopedia. This arose when I was reading the ...
DannyNiu's user avatar
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1 vote
0 answers
28 views

Calculating expectation by manipulating into the MGF

Given that the moment generating function of the random variable X is $M_X(t)=\frac{1}{1-2t}$, calculate the expectation of $Y^n,n\in \mathbb{N} $, where $Y=365 \times 0.3^X$. Here is my attempt. Use ...
Starlight's user avatar
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0 votes
1 answer
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Property of mean preserving spread

Consider two random variables $X\sim F$ and $Y\sim G$, both on $[0,1]$. Suppose $X$ is a mean preserving spread of $Y$, i.e., $$ \int_t^1 F(x)dx \le \int_t^1 G(x)dx $$ for all $t\in[0,1]$ and equality ...
kenji's user avatar
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1 vote
2 answers
44 views

Linearity of expectation for infinite sums of positive random variables

I have a question regarding the linearity of the expectation of the infinite sum of positive random variables, namely I want to prove on a countable space $\Omega$ that \begin{equation} \mathbb{E}[\...
Leoncino's user avatar
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2 votes
0 answers
77 views

Expectation of Bernoulli random variables and independence

Suppose $X_1,X_2,\dots,X_n$ be a sequence of Bernoulli random variables satisfying $$\mathbb E[X_{i_1}X_{i_2}\dots X_{i_k}]=\mathbb E[X_{i_1}]\mathbb E[X_{i_2}]\dots \mathbb E[X_{i_k}]$$ for any $2\...
maths and chess's user avatar
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52 views

Role of the PDF in the conditional expected value of a continuous random variable

I'm looking to develop a more accurate understanding of the role the pdf plays in the conditional expected value of a continuous random variable. I have previously only calculated this for random ...
Hal_Incandenza's user avatar
0 votes
6 answers
120 views

Revisit: A fair die is rolled until 3 (consecutive) 4's appear: what is the expected # of 2's and 3's?

This is a self-answer question that revisits this low quality posting. MathSE answers have an upper bound on the size of this answer. So, I split the self-answer into two parts, Part-1 and Part-2. As ...
user2661923's user avatar
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2 votes
1 answer
59 views

Question about notation in Durrett's book

I have been studying Markov Chains through Rick Durrett's book, more precisely I was focusing on the Markov Property (Theorem 5.2.3 on page 276 of the book available at: https://services.math.duke.edu/...
Monteiro_C's user avatar
3 votes
2 answers
40 views

If $f(x)$ is a pdf that is symmetric about $a$ (AKA $f(a+x) = f(a-x)$ for all $x \in \mathbb{R})$ then $EX = a$.

I'm looking for help for the problem in the title. If $f(x)$ is a probability density function (so it's nonnegative and its integral from $-\infty$ to $\infty$ is $1$) and is symmetric about $a$ (EDIT:...
S.H.'s user avatar
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0 votes
0 answers
40 views

Notation of Expected value, when game with loss function and strategy as variable.

I have the case where I need to write the expected value of a score $\mathbb{E}[S]$ in a game where we can use different strategies $\sigma$ and also the loss function $f$ may change, which I write ...
Ziur Olpa's user avatar
  • 101
4 votes
2 answers
199 views

Order given by two probability measures

Let $X$ be a r.v. and two measures $\mathbb{P}$ and \mathbb{Q} such that \begin{equation} \mathbb{E}^\mathbb{P}[X]\leq \mathbb{E}^\mathbb{Q}[X]\quad (1) \end{equation} It implies \begin{equation} \...
Don P.'s user avatar
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5 votes
1 answer
238 views

$X$ and $Y$ are i.i.d. then $\mathbb{E}(|X|) \le \mathbb{E}(|X+Y|)$

I don't know how to prove this: If $X$ and $Y$ are i.i.d. then $\mathbb{E}(|X|) \le \mathbb{E}(|X+Y|)$ I first want to use conditional expectation. When $\mathbb{E}(X)=0$, I know it is right. $\...
Yu GongLian's user avatar
0 votes
0 answers
39 views

Help with probability problem involving tourist and bear encounters

I am trying to solve the problem below but having a hard time doing it, any help appreciated (my attempt at the bottom). Specifically, I do not know how to solve (b) and I am unsure whether I got ...
Tymofii256's user avatar
2 votes
0 answers
73 views

Expectation of max of sums of independent r.v.s

Let $\{ X_{1}, X_{2}, Y_{1}, Y_{2} \}$ be independent with $X_{i} =^{d} Y_{i}$ for $i \in \{ 1, 2\}$ and $\mathbb{E}(X_{1}), \mathbb{E}(X_{2}) < \infty$. Is it true that: $$\mathbb{E}(\max\{(X_{1}+...
Snidd's user avatar
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0 votes
1 answer
72 views

Find $\mathbb{E}[\det(A-A^T)]$ if $a_{ij} \sim \text{Bernoulli}(p)$ [closed]

Let $A$ be a random matrix, which elements take values $1$ and $0$ with probably $p$ and $(1-p)$ respectively. What is $\mathbb{E}[det(A-A^T)]$? When dimensionality of $A$ is odd, $det(A-A^T) = 0$, as ...
innerproduct's user avatar
0 votes
1 answer
28 views

Finding Expectation of a Uniform random variable from its moment generating function

From Taylor series, if I need to get the k-th moment, I need to find the k-th derivative of the Moment Generating function. If I have $X \sim \text{Uniform}(0,1)$, the MGF $M_{X}(s)$ is; $$M_{X}(s) = \...
moseskabungo's user avatar
2 votes
1 answer
61 views

An application of Chebyshev association inequality?

Let $X$ be a r.v and let $f \geq 0 $ be a nonincreasing function, $g$ be a nondecreasing real-valued function. Suppose $h\geq 0$ is a function such that $h(X)$ has finite expectation with $E[h(X)f(X)]...
ProbabilityLearner's user avatar
1 vote
1 answer
69 views

Car parking - quant guide

Suppose that we have 2 cars parked in a line occupying spaces 1 and 2 of a parking lot. Spaces 3 and 4 are initially empty. Every minute, a car is considered eligible to move forward one space if a) ...
bigstreet's user avatar
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0 votes
1 answer
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Clarification regarding betting game and linearity of expectation

Here is the problem I'm dealing with: You have 3 blue and 3 red cards. These cards are mixed and placed face-down in a deck, ready to be turned over one-by-one. Before each card is turned, you are ...
Abhay Agarwal's user avatar
0 votes
1 answer
51 views

Mutual information expansion not justifiable

I have recently read a mutual information term, $I(X;Y,Z)=E_{p(X,Y,Z)}\big[\log\frac{ p(X|Y,Z)}{p(X)}\big]$. While this expansion does not make sense to me. Is it correct? My understanding (using ...
Aleph's user avatar
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0 votes
0 answers
52 views

Max of sums of independent elements in two n-tuples

Let $X_{1}, ..., X_{n}, Y_{1}, ..., Y_{n}$ be independent random variables with finite expectation. Assume that for $1 \leq i \leq n$, it is the case that $X_{i}$ is equal in distribution to $Y_{i}$ (...
Snidd's user avatar
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0 votes
0 answers
19 views

Existence of $E(u)$; $E(u)=E(u^+-u^-)$

From chapter $IV.2$ of $$\textit{William Feller 'An Introduction to Probability Theory and Its Applications'; Vol.2}$$ $E(u)=\int_{\mathcal R^r}u(x)F\{dx\}$ I'm confused about the statement '$E(u)$...
J P's user avatar
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0 votes
0 answers
26 views

expected value of the Euclidean norm for a multivariate non-zero mean normal vector calculation

I found this general expresson. Could anyone assess it's correctness ? The expected value of the Euclidean norm for a multivariate normal vector $$(\mathbf{K} \sim \mathcal{N}(\mathbf{u}, \Sigma))$$ ...
Michael P's user avatar
2 votes
0 answers
38 views

Expectation of bessel process conditioned on starting point

I want to calculate the expectation of the Bessel process for $n=3$ $$\begin{align} dX_t = dW_t + \frac{1}{X_t} dt \end{align} $$ given that we start at initial point $X_0=1$. My attempt was the ...
black's user avatar
  • 279
0 votes
0 answers
21 views

Ergodicity when the variance doesn't converge

A power law (x^(-k)) isn't ergodic for k<= 2 given that the expected value doesn't exist (nothing to converge to). However for 2<k<=3 there is a finite expected value but an infinite variance....
David's user avatar
  • 37
0 votes
1 answer
48 views

Determine probability distribution by expected value

Suppose we have a discrete random variable $X$ taking values in $\mathscr{X}:=\{x_1<x_2<\cdots<x_n\}\subset\mathbb{R}$. Given some probability distribution $\mathbb{P}$, we have the expected ...
Matt's user avatar
  • 893
2 votes
2 answers
99 views

Random walk on the edges of a square where staying at same position is allowed

The classic question of finding the expected number of steps to move from one corner of a square to the opposite. I found an intuitive way to do it by considering that we are at position $(0,0)$ and ...
Axo's user avatar
  • 301
2 votes
1 answer
39 views

Excess waiting time given two exponential variables

Suppose that there are two patients that arrive on time to a clinic. The time that each patient takes with the doctor is distributed according to an exponential and the expectation of that is 0.5 ...
One's user avatar
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