Questions tagged [expected-value]

Questions about the expected value of a random variable.

Filter by
Sorted by
Tagged with
0 votes
0 answers
11 views

Expectation of the Estimated Slope

In the following book, the expectation of the estimated slope with linear regression is derived as follows: $$ \begin{align*} \hat{\beta}_1 ~ &= ~ \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(Y_i ...
user avatar
0 votes
0 answers
21 views

Question on merging Expected values

For my course on financial mathematics i have the following exercise (with solution): What I do not get about this solution is how they get $\xi$ to turn up everywhere. For example, the merge $E[Y|G] ...
user avatar
0 votes
1 answer
33 views

Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Suppose that $X(t)$ satisfies $\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7.$ Find the mean and the variance of $X(1).$ I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
user avatar
  • 359
0 votes
1 answer
27 views

A probability of the power of a random variable, but inside an integral

In "Concentration inequalities - A nonasymptotic theory of independence" by Boucheron et al. I found this weird statement I can't quite understand. It says, for a random variable $X$ which ...
user avatar
1 vote
0 answers
21 views

Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
user avatar
  • 1,047
0 votes
1 answer
33 views

Finding $E(XY)$ from a uniform distribution

I have this question where I got $E(X)$ already but I'm struggling with $E(XY)$: Let $U$ be a random variable uniformly distributed over $[ 0,2π ]$ . Define $X = \cos U$ and $Y = \sin U$. Show that $X$ ...
user avatar
0 votes
1 answer
26 views

Finding mean of sum where there's no definite stop

I'm having trouble with this question: Suppose $r$ chips are grabbed from $n$ chips where each chip is numbered $1$ to $n$. Let V be the sum of the chips. Find E(V) What I thought of so far was the ...
user avatar
-2 votes
0 answers
20 views

expected value for $Y$ [closed]

We have generated $x_1,x_2,x_3 \dots x_n \sim Ber(0.5)$, after we generate $y_1,y_2,y_3 \dots y_n \sim N(1-2X,1)$ how to calculate $E(Y)?$
user avatar
  • 115
2 votes
1 answer
39 views

Expected value of maximum of two numbers, where one is normal distributed

I am searching something similar to the first order loss function ( $\mathbb{E}[max({y_{i}-y_{i},0})])$ (where $y_{i}$ is normally distributed) but for $\mathbb{E}[min(y_{i},d_{i})]$ , where $y_{i}$ ...
user avatar
-4 votes
0 answers
28 views

Statement is true or false? [closed]

Prove the true ones and give counterexamples for the false ones. Let $X$ and $Z$ be random variables. If X and Z are independent, then $E[X^2] = E[Z^2]$.The statement is true or false? If true or ...
user avatar
6 votes
3 answers
316 views

Expected length of sum of vectors

Suppose we have $n$ arbitrary unit vectors $v_1, v_2, v_3, \dots, v_n$. (Here, a "random" unit vector is defined as $\langle \cos(x), \sin(x) \rangle$ for a random $x$ such that $0 \leq x &...
user avatar
-1 votes
0 answers
21 views

Calculate the expectation and variance for normal distribution [closed]

I am solving a question which is about distribution but it is confusing as I am new in the course of applied data analysis. how we calculate the expectation and variance when we have this given ...
user avatar
0 votes
2 answers
19 views

Square Integrable random variables are always integrable?

If $X$ is any random variable on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{E}(X^2)<\infty$, how can I show that $\mathbb{E}(|X|)<\infty$, ie that $X$ is integrable? I'm ...
user avatar
3 votes
1 answer
98 views

Expected value suggested by simulation

If $X$ is random variable giving the first $n$ value such as $X_1+\cdots+X_n$ is multiple of $m \in \mathbb{N}^*$, where $X_i$ are independent variables with uniform distributions on $\{1,\cdots,k\}$. ...
user avatar
  • 221
4 votes
1 answer
85 views
+50

Finding the expected stopping time of random walk

been stuck on this optional stopping theorem quesiton for some time. I feel like I'm heading in the right direction but I am not sure if I am right. Problem Consider a random walk $\left(S_{n}\right)_{...
user avatar
6 votes
1 answer
67 views

Surprising appearance of triangular numbers during simple coin flip game

I have the following problem: You have $n$ coins in a row in some beginning state of heads/tails. Define a process as follows: If you have $k>0$ heads, flip over the coin in the $k$th position ...
user avatar
0 votes
0 answers
25 views

what is the expectation of identity matrix

I would like to get an upper bound for this term $$ E\left[\left( \|I\| +b\right)^2 \left\|c^2\|x-x_1\|\right\|^2\right]$$ where $I$ is the identity matrix and b and c are constants. Am I allowed to ...
user avatar
4 votes
1 answer
49 views

Convergence of non iid observations on the empirical distribution

Let $f$ be a function on domain $X$ with binary output $f: X\to \{0,1\}$. We define an arbiatry distribution $\mathcal{Q}$ over $X$ and the empircal distribution of $n$ samples from $\mathcal{Q}$ -- $\...
user avatar
0 votes
0 answers
33 views

What is the conditional covariance/expectation of two independent random Poisson variables?

Say I have two random variables $Z^{p}_{i} \mid Y^{p}_{i} \sim \text{Poisson} \left(t^{p}_{i}Y^{p}_{i}\right)$ and $Z^{q}_{i} \mid Y^{q}_{i} \sim \text{Poisson} \left(t^{q}_{i}Y^{q}_{i}\right)$ for ...
user avatar
-2 votes
1 answer
49 views

Conditional expected mean

I’m stucked on the following problem, and i just can’t seem to grasp what to. $$Let \: X_{1}, X_{2},… \: be\: i.i.d. \: random \: variable, \: and \: assume\: that: \\ \mathbb{P}(X_{1} =1)=1-\mathbb{P}...
user avatar
  • 1
2 votes
1 answer
27 views

Problem in finding expectation of two consecutive faces in consecutive throws of a die

I came up with a question while doing a project on simulation on some experiments regarding dice. The question was: On average, how many times must a 6-sided die be rolled until two rolls in a row ...
user avatar
  • 579
0 votes
0 answers
13 views

Integrating Conditional Probabilities

I have a function : F(a,b,c) I want to take the Expected Value of this function in the following form: E[ F(a,b,c) | a, c) I know that for a single variable, the Expected Value is E(x) = integral(x*p(...
user avatar
0 votes
0 answers
27 views

What is $\mathbb{E} \bigg\lvert \int f(x)\Pi(dx)\bigg\rvert^2$ for a random measure $\Pi$?

As mentioned in this wiki page we know that $$\mathbb{E}\int f(x)\Pi(dx)=\int f(x)\mathbb{E}\Pi(dx)=\int f(x)\nu(dx),$$ where $f$ is any measurable function and $\Pi$ is a random measure with ...
user avatar
  • 1,190
-2 votes
0 answers
32 views

Bound on the probability of the size of a random walk [closed]

Let $S_n=X_1+\dots+X_n$, where $X_1,X_2,\dots$ are independent with $\mathbb{E}(X_m)=0$ and $\mathbb{E}(X_m^2)=\sigma_m^2\in(0,\infty)$ for all $m\geq1$. Let $\mathcal{F}_n=\sigma(X_1,\dots,X_n)$. I ...
user avatar
0 votes
0 answers
7 views

coefficient of kurtosis in higher moments of random variables

I don't quite understand why the $-3$ from the $\gamma_2$ equatioon is not used in the answer for the question below. It's even mentioned in the answer too! 3.6.21. Calculate the coefficient of ...
user avatar
0 votes
0 answers
26 views

Question about derivative of an Expected Value

Let $Z_0(t) \approx V_0e^{\lambda_0 t}$ be a branching process where $V_0 \sim \text{Exp}(\lambda_0/a_0)$, $\lambda_0 = a_0-b_0$ and $a_0,b_0 \in \mathbb{R+}$. I do not understand one step from this ...
user avatar
2 votes
1 answer
29 views

Question on Conditional Expectation. with sum of Bernouli random variables.

X is a uniform random variable on $[n]=\{1,2,\dots ,n\}$ and for every $i\in[n]$ We define$ Y_i \sim Ber(\frac{1}{2})$ ,such that all The variables are independent(all The Y's and X). now we have to ...
user avatar
  • 99
0 votes
1 answer
14 views

Why are these equations describing the variance of residuals equivalent (single-factor model in finance)? [closed]

Residual Equations The above equations describe the residual variance in the single-factor model in finance. I'm struggling to understand why they are equivalent, specifically why the fourth and fifth ...
user avatar
1 vote
2 answers
25 views

Inequality involving moments of a distribution [closed]

Let X be a real random variable. Under what conditions on the distribution do we have that $$\mathbb{E}( X^{2n + 2}) \geq \mathbb{E}( X^{2n}) \mathbb{E}( X^{2})$$ for all integer $n$? I tried using ...
user avatar
  • 19
0 votes
1 answer
24 views

Probability mass function and expected value in lottery

I can buy one lottery ticket out of two available. In the first lottery I can win \$100 with probability 0.1, and the price of ticket is 10. In the second lottery, I can win \$50 with probability 0.1 ...
user avatar
  • 33
1 vote
1 answer
37 views

Taking expectation in proof of Markov inequality

In Probability and Computing (Mitzenmacher & Upfal), Markov inequality is proven roughly as follows: (Theorem) Markov's Inequality. Let $X$ be a random variable that assumes only nonnegative ...
user avatar
  • 141
-1 votes
0 answers
52 views

On the expectation of conditional expectation [duplicate]

Let $X,Y,Z$ be real, integrable, continuous random variables. It is known that $E[E[X|Y,Z]]=E[X]$. What happens if we take the "expectation" on only one of the conditioning random variables? ...
user avatar
  • 2,038
2 votes
1 answer
76 views

Exchanging derivatives and integrals with the delta function

Let $H(x) = \begin{cases} 1 \text{ if } x \geq 0 \\ 0 \text{ if } x < 0 \end{cases},$ and let $\delta(x)$ be its distributional derivative (the Dirac Delta function). I have a function of the form $...
user avatar
  • 557
1 vote
1 answer
47 views

calculation of $E[\Phi(X)]$

Let, $X\sim N(\mu, \sigma^2)$ and let $\Phi(\cdot):\mathbb{R}\to[0,1]$ be the CDF of a standard normal distribution. Then, what is the pdf of $Y=\Phi(X)$. Also, find $E[\Phi(X)]$. Note:- Here, $Y=\...
user avatar
  • 790
2 votes
0 answers
18 views

Expected Value of Slot Machine

I have a probability question that has to do with slot machines. Here is how the game works: There are two reels, one on the left and one on the right. There are several symbols on each reel, one of ...
user avatar
  • 79
0 votes
0 answers
53 views

How to find the expected value of an exponential curve?

I have a function of the form $$ y = \left(\frac{a}{b}\right)^{1/T} - 1$$ where $T$ is a future point in time, such that $$0 < T \leq \infty.$$ Question 1: Given constant values for $a$ and $b$, ex:...
user avatar
  • 101
0 votes
2 answers
42 views

Expected time until k failures among n items each independent and having exponential distribution

This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution ...
user avatar
0 votes
0 answers
23 views

Deriving the expected value of a random variable via another random variable

Let $T = \{1,2,\ldots\}$ and $T_0 = \{0\} \cup T$ denote finite sets. Further let $X_t$ denote a random variable with PDF $f$. Let $x = (x_t)_{t \in T}$ denote a vector of realizations of $(X_t)_{t \...
user avatar
  • 732
0 votes
0 answers
30 views

Expect return of a standard Brownian motion

I'm working on the problem for a university project, I struggling for several days, I would post the same question with different coefficients from the problem I currently working on, so please show ...
user avatar
0 votes
1 answer
70 views

I need to prove that $E(X+Y) = E(X) + E(Y)$ for discrete random variables but just using the definition of expected value

First of all i want to say that i know (and i checked) other posts that talks of the same subject, but i'm trying a different perspective, and that's why i'm writing this topic. Well, i need to prove ...
user avatar
1 vote
1 answer
59 views

How do I prove that the expectation value of the following two random variables converge to zero?

I have given $X_n,Y_n$ two sequences of real valued random variables in the same probability space. I assume that $X_n\Rightarrow X$ in distribution and $|X_n-Y_n|\rightarrow 0$ in probability. In ...
user avatar
  • 1,029
1 vote
1 answer
59 views

Is the difference of random walk a martingale

Suppose $X_i \sim i.i.d. N(0,1), i=1,...$, $S_n=X_1+...+X_n$ let's $ Y_{i}^{\left( v \right)} := X_i \textbf{1}_{ \left\{ \left| X_i \right| \leq v \right\} }, X_{i}^{\left( v \right)} := \left( Y_{i}...
user avatar
  • 25
4 votes
2 answers
77 views

Question about conditional expectation fact

This fact comes from *Concentration of Measure for the Analysis of Randomised Algorithms * by Dubhashi and Panconesi (page 76, equation 5.2). Let $X$ and $Y$ be two discrete random variables and two ...
user avatar
  • 593
2 votes
1 answer
38 views

Surprisingly simple expected time for the "range" of a Brownian motion to extend beyond $a$ - is there a martingale method?

I will call the range $R(t)$ of a standard Brownian motion the difference between its maximum $M(t) = \max_{0\leq s \leq t} B(t)$ and its minimum $m(t) = \min_{0\leq s \leq t} B(t)$. That is, I am ...
user avatar
2 votes
2 answers
55 views

Show that $\mathbb{E}\Big[\mathrm{ln}\big(\frac{X}{\mathbb{E}[X]}\big)\Big] < 0$

We know that $X > 0$ and that $\mathbb{E}[X] < \infty$. Show that $\mathbb{E}\Big[\mathrm{ln}\big(\frac{X}{\mathbb{E}[X]}\big)\Big] < 0$. Could someone show me a way to prove it? Thanks.
user avatar
0 votes
1 answer
18 views

Expected value as well as variance of observing a TH string in coin flips

I'm working with a problem where I need to calculate the expected value and the variance of the amount of coin flips needed until we observe a a "TH" - string in a coin flip experiment. In ...
user avatar
  • 1,450
3 votes
1 answer
37 views

Showing that a function of two brownian motions is a martingale.

Let $B$ be a standard Brownian motion, let $f$ be a smooth function taking values in $[a,b]$ where $0<a<b<\infty$ and assume that the derivative $f^\prime$ is bounded. For $t\in[0,1]$ and $x\...
user avatar
1 vote
1 answer
53 views

A “two-dimensional” generalization of coupon collectors problem

Suppose we have colored balls with numbers. There are $m$ colors and $n$ numbers (so there are $mn$ different kind of balls, appearing with equal probability). What’s the expectation of balls we need ...
user avatar
  • 219
5 votes
1 answer
56 views

Probability of stopping time being finite.

Let $X$ be a continuous non-negative local martingale with $X_0=1$ and $X_t\to0$ almost surely as $t\to\infty$. For $a>1$, let $\tau_a=\inf\{t\geq0:X_t>a\}$. I am tasked with showing that $\...
user avatar
6 votes
2 answers
98 views

Flipping $n$ coins until they're all heads

Suppose you have $n$ (fair) coins: You flip them until they're all heads in the following sense: Suppose you flip all of them first and get $k$ heads. This is round $1$ You remove these heads from the ...
user avatar

1
2 3 4 5
182