Questions tagged [expected-value]

Questions about the expected value of a random variable.

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20 views

Expected value of coin game - double money on heads

We have $1 on the table and a quarter. Every time we flip the coin and it lands on heads, the money on the table doubles but once it lands on tails, the money on the table goes to 0 and the game ends. ...
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Finite-variation processes coupled by geometric brownian motion

We have a geometric Brownian motion $$dS(u) = \mu S(u) dt +\sigma S(u) dZ(u), \quad S(0)=1,$$ where $Z$ is a standard Brownian motion. There are two nondecreasing continuous processes $L$ and $U$ with ...
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Expected value question solution

We flip 2 coins:Our random value is the number of heads Possible values: $$\begin{array}{|c|c|c|c|}\hline &TT& HT ~~~ TH & HH\\\hline X& 0& 1& 2\\ \hline p(X=...
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2answers
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Expectation of a equation with uniformly distributed variable

There is the following problem: One maximises profit (v-0.5)x, where x ∈ {0,1} and v~U(0,1). Now, in order to maximise this one chooses x=0 if v<0.5, and x=1 if v≥0.5. Now, one has to find the ...
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Expected length of stick broken at two points using calculus

This question is from 50 problems in probability. Q. A glass rod is coloured red at one end and blue at other. Many such rods got broken into 3 parts. What is the average length of part colored blue ...
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1answer
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What is wrong with my approach for CLRS 5.4-6 : Given n balls and n bins,find expected number of empty bins?

I am trying to find expected number of empty bins after n balls are tossed into n bins. And each toss is independent and equally likely to end up in any bin. Below is my approach. My indicator ...
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5answers
250 views

Two different books are giving two different solutions.

So I am solving some probability/finance books and I've gone through two similar problems that conflict in their answers. Paul Wilmott The first book is Paul Wilmott's Frequently Asked Questions in ...
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3answers
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Expected number of cards in original position in a shuffled deck of $52$ cards?

Assume is shuffle is quite good that it randomizes the card order. We know that E = $ \sum_{X=1}^n X*P(X) $ We are already know that n=52 and that there are 52! ways to arrange the cards. So ...
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28 views

What is the Covariance of Two Bernoulli Random Variables coming from the same sample, under design-based approach?

The scenario for the question is as follows: We draw a sample of size n from a population of size N. Assume a design-based approach (meaning the variable of interest $y_i$ (where $i$ represent the $i^{...
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3answers
56 views

Trader with 60% chance of gaining 50% and 40% and 40% chance of dropping 50%. Average return per day calculation.

I am reading Paul Wilmott's Frequently Asked Questions in Quantitative Finance, and there is a question that states the following: Every day a trader either makes 50% with prob- ability 0.6 or loses ...
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How many times must a bounded random number generator bounded by its output run until it outputs 1?

Say you have a function with input $x$ that generates a random integer between $1$ and $x$. Every time you run the function you replace its input with its output until the function outputs $1$. What ...
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Expectation of product of geometric Brownian motion

I have a geometric Brownian motion as: $$\frac{d x_t}{x_t}=\mu_t dt+\sigma_t dZ_t$$ I need to find the expectation of the product at two different points in time to some power $\chi$, conditional on ...
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Variance and Expectation of Little o

Let $X$ be a random variable dependend on some parameter $t$. with finite expectation and variance for all $t$. How can I compute $E[X(t) + o(t)]$ and $V[X(t) + o(t)]$ for $\rightarrow 0$? From ...
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1answer
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The expected value $E(Z-XY)$ when $f_{x,y,z}=24x$ when $0<x<y<z<1$ and 0 otherwise

I've been trying to find the expected value $E(Z-XY)$ when $f_{x,y,z}=24x$ for $0<x<y<z<1$ and 0 otherwise I've tried to extract the density function for each variable, and then calculate $...
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3answers
55 views

Compute $ \int_{\mathbb R} |\sin x| e^{-(x-a)^2 / b^2} dx $

For $a, b > 0$, I want to compute $$\int_{\mathbb R} |\sin x| e^{-(x-a)^2 / b^2} dx.$$ What I have tried: Since $\int_{\mathbb R} |\sin x| e^{-(x-a)^2 / b^2} dx= \sum_{k \in \mathbb Z} \int_{0}^{\...
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1answer
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Optimizing expectation, unknown parameters, normal distribution. How many restaurants should I try before choosing one for the rest of my n-m meals?

Suppose you move to a new city with an infinite number of restaurants and you plan to stay there for a predetermined amount of time. You plan to have n meals at restaurants over the course of your ...
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2answers
30 views

Expected payout after a deductible

To my understanding E[X-1] = E[x] - 1. So why doesn't the following work? A policy has a deductible of 1 and a capped payment of 5 and losses are distributed exponentially with mean 2. The expected ...
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1answer
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Given $K * N$ grid such that $L$ spots on the grid are filled, what is the expected number of columns with at least one spot filled?

This seems to be a simple question but I can't seem to wrap my head around it. let's say we throw $L$ marbles onto a $K*N$ grid, such that each spot in the grid can hold at most one marble. how many ...
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1answer
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Find the expectation of a parameter estimate given a sample [closed]

Given a sample $y$, drawn from a binomial distribution with an unknown parameter $p$, find the expected value of $p$. I believe the answer would be $\frac{\sum y_i}{n}$ (that is the MLE) but not sure ...
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finding constant in the bound for the geometric random variable

It was shown here Moments of Geometric Random Variable that if $X$ is a geometric random variable, i.e. it represents the number of consecutive failures before you get the first success, where the ...
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1answer
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Does the mathematical expectation inequality $E\{g(V)\}<c$ hold?

Assume that $V$ is a stochastic variable, $g(V)\geq 0$ is a function related to $V$. If the upper bound of $g(V)$ can be determined, i.e. $g(V)<c$, where $c$ is a constant, does the mathematical ...
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Proving a bound on the difference between expected value of a continuous random variable and the expected value sampled on all integers

I need to prove that for a random variable $X$ with $$ X \geqslant 0 $$ it's true that $$ \sum_{n=1}^{\infty} P(X \geqslant n) \leqslant E[X] \leqslant 1 + \sum_{n=1}^{\infty} P(X \geqslant n) $$ ...
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35 views

Expected value of number of families? [closed]

I have 100 different families in a community, each family with 10 members. If I select 100 members from the community at random, what is the expected value of the number of different families they ...
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1answer
42 views

Is it possible to determine the probabilities with this information?

I play a game where there are loot boxes that follow a probability distribution. In a loot box, you always have 3 loot pieces. The following information is given: Chance of ATLEAST 1: 100% - Rare ...
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1answer
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Solve transformation of expected value

I try to solve for $a > 0$ the integral $$\int_\mathbb{R} e^{ax} f(x) \,dx$$ where $f$ is the density of a $\mathcal{N}(0,1)$-distributed random variable (w.r.t. the Lebesgue measure) using the ...
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Archimedean Copula Probabilities

I have the following excercise : Let C be an Archimedean copula with generator given by $\psi(x) = E[e^{ −xV} ]$, where V is an exponentially distributed random variable with expectation 1. Calculate ...
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1answer
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Expected value of a deterministic function defined on a random domain

I'm curious of how to find the expected value of the following. Suppose we have a constant function defined on, $$f(x) = \begin{cases} 1 & if \ \ \ 0 < x< D \\ ...
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Probabilistic Recurrence Relations

In the paper "The Complexity of Parallel Search" (Karp, Upfal and Wigderson; Journal of Computer and System Sciences 36, 225 - 253, 1988) in the appendix, the authors present the iteration ...
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2answers
47 views

Absolute moment existence

I have given $$F_X(x) = \frac{1}{1+e^{-x}} ; \quad x \geq0$$ I have to calculate $k$, s.t. $E[|X|^k]<\infty$ I did calculate the density by $f_X(x) = \frac{dF}{dx} =\frac{e^{-x}}{(1+e^{-x})^2} $ . ...
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1answer
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Expected number of tosses to get 3 heads (not necessarily in a row)

Intuition tells me the expected number of tosses of a fair coin needed to get 3 TOTAL heads is 6. I'm trying to show this using purely (1) Bernoulli random variables and (2) purely Geometric random ...
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1answer
34 views

Compute average volume for a normal distribution of particle diameters [closed]

Given a normal distribution of particles with mean diameter m and standard deviation s, how can one compute the volume average ...
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1answer
42 views

Second Moment of shifted Geometric distribution

I know that the PGF of a shifted geometric distribution is $\frac{zp}{1-z(1-p)}$ where $z$ is the observed value of the random variable $Z$ and $0<p<1$. I am facing difficulties finding $G^{''}...
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The expectation and variance of the $\infty$-norm $\|\hat{\mu}-\mu\|_{\infty}$. [closed]

Suppose the i.i.d. sample $\{X_i\}_{i=1}^n$ follows a $p$-dimensional normal distribution $N_p(\mu, \Sigma)$, where $\Sigma>0$ and the eigenvalues of $\Sigma$ is bounded away from $0$ and $\infty$. ...
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1answer
51 views

Calculating individual mean from a sum of random variables.

Suppose that I have $X_1, X_2, X_3$ as a sequence of independent random variables with $E(X_i)<+\infty$ but not identically distributed, so they have different $E(X_1)\neq E(X_2)\neq E(X_3)$. Let's ...
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1answer
32 views

Calculating the expected value of clients served by two shops

I need some help to solve this problem: There are two shops and each can serve at most one client per period. The number of clients that arrive in each period to each shop is \begin{cases} 0 &...
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Computing $E_{\mu_{k}, \Lambda_{k}}\left[ (\mathbf{x}_{n} - \mu_{k})^{T} \Lambda_{k}(\mathbf{x}_{n} - \mu_{k}) \right]$

I wish to find: $$ E_{\mu_{k}, \Lambda_{k}}\left[ (\mathbf{x}_{n} - \mu_{k})^{T} \Lambda_{k}(\mathbf{x}_{n} - \mu_{k}) \right] $$ where the pdfs are: $$ f( \mu_{k} , \Lambda_{k}) = f( \mu_{k} | \...
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1answer
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Expected waiting time in strings given by derivative of Generating Function

This is exercise I.33 from Analytic Combinatorics by Flajolet & Sedgewick, with the important bit in bold: Waiting times in strings. Let $\mathcal L \subset \mathrm{SEQ}\{a, b\}$ be a language ...
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Prove that $E[N_r^2] = u_1 + \cdots + u_r + 2 \sum_{j=1}^{r-1} u_j(u_1 + \cdots + u_{r-j}).$

(Feller Vol.1, P.340, Q.20) Let the recurrent event $\varepsilon$ and $N_r$ the number of occurrences of $\varepsilon$ in $r$ trials. Prove that $$E[N_r^2] = u_1 + \cdots + u_r + 2 \sum_{j=1}^{r-1} ...
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1answer
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Persistent Markov Chain with infinite mean recurrence time

Given a Markov Chain $(X_n)_{n \ge 0}$, state $i \in S$ is defined as persistent if $P(T_i < \infty | X_0 = i) = 1$ (where $T_i$ is the first passage time to state $i$). Moreover, the mean ...
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1answer
40 views

Expected value of a variable with piecewise pdf

If I have that the pdf of a variable is: $$f_Y(y) = \begin{cases} \frac{1}{2}, & \text{if $y=0$} \\ \frac{1}{4a}, & \text{if $a<y<2a$} \\ \frac{1}{4a}, & \text{if 4a<y<5a}\\ ...
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Normalized Gaussian vector: correlation between components?

Let $X\sim\mathcal{N}(0,C)$, where $C\in\mathbb{R}^{N\times N}$ is the covariance matrix, and define $Z$ by normalizing $X$ with its Euclidean norm $Z:= X/\|X\|_2$, i.e., $Z_i:=\frac{X_i}{\sqrt{\sum_{...
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1answer
19 views

Using expected value of number of attendees to an event to calculate expected revenue

In the problem you pre-sell 21 non-refundable tickets values at 50 dollars to an event and can only accommodate 20 people. But in the event the 21st person shows up you must pay the person who's out $...
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1answer
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Probability and combinatorics - Find the expectancy of people in a line and in circle

$n$ men and $m$ women are standing in a line (randomly). Find the expectancy of the number of men that stand beside a women (at least one side - left or right) Harder question: Now solve it, but they ...
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1answer
35 views

Expectation of independent random process

Let $[v_1(k) \, v_2(k)]^T\in\mathbb{R}^2$ be a stochastic process that is independent of the process $\beta_i(k)$ for all $i=1,2,3,4$, with $\mathbb{E}[\beta_i(k)]=\bar\beta_i $. Both of them are ...
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33 views

Finding the distribution of a vector of random variables

I've encountered this question: Let (X,Y) be a jointly distributed normal vector. Find the distribution of (X, E(X|Y)) assuming Cov(X,Y) isn't singular. I'm unsure what to do - I tried using the joint ...
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2answers
43 views

Double differentiation of characteristic function of Normal random variable

Knowing that a one-dimensional random variable $\Gamma$ is Gaussian if it has the characteristic function $$\mathbb{E}\hspace{0.15cm}e^{i\xi\Gamma}=e^{im\xi-\frac{1}{2}\sigma^2\xi^2}\tag{1}$$ for some ...
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1answer
18 views

Why is the expectation of the random variable “W” equal to the limited integral of the survival function of the random variable “Y”?

Ok, so here's the problem. Let the random variable X represent loss in 2005. The density function of X is exponential with mean equal to 1. Let the random variable Y represent loss in 2008. Y = 1.2X ...
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69 views

Expected Value Bounded by Median - a Direct Proof?

Let $f:[0,1]\to \mathbb{R}$ be a positive, continuous and strictly increasing function such that $$\int_0^1 f(x)dx=1$$ Let $\xi\in (0,1)$ be such that $$\int_0^{\xi}f(x)dx=\int_{\xi}^1f(x)dx=\frac{1}{...
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1answer
31 views

Does $E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0$?

$$E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0\,?$$ I am wondering if the above statement holds true. The LHS is saying that given any value of $y_i$, the expected value of $x_i$ is ...
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43 views

Ratio of the expectation over the joint distribution and the expectation over the marginal distribution

I have the following expression and I am having trouble understanding it $$\frac{\int y f_{Y,X}(y,x) dy}{\int y f_Y(y) dy}$$ Where $f_{Y,X}(y,x)$ is the joint density function of random variables $X$ ...

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