For questions about the expectation of a random variable: computations, upper/below bounds, etc...

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-4
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0answers
16 views

Expected value of adjacent bulbs go off [on hold]

Given $n$ light bulbs with exponential distribution for the probability when the bulb will be turned off, what is the expected time that any two adjacent bulb will be off?
0
votes
1answer
34 views

Expected number of bins with more than one ball

Suppose that $n$ balls are randomly thrown into $N$ bins. We can compute the expected number of bins that contain at least one ball as $E(X) = N(1 - (1 - 1/N)^n)$. Now, suppose that instead we are ...
3
votes
4answers
207 views

$\int_{0}^{\infty}xe^{-x^2/2}dx= 1$?

$X \sim N(0, 1)$ $$E(|X|) = \frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}|x|e^{-x^2/2}dx= \frac{2}{\sqrt{2\pi}}\int_{0}^{\infty}xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}$$ I don't understand how the last ...
6
votes
3answers
151 views

Paradoxical Game Show Problem [duplicate]

Here's a problem that has had me scratching my head for a long time: Imagine you're in a game show, and are presented with 2 boxes. You are told that both boxes contain a sum of cash, but one of the ...
0
votes
1answer
36 views

Expected value of the floor function of a sum of two variables

In a recently published paper I have encountered the following equality. Let $U$ be a random variable uniformly distributed in $[0,1]$ and let $Z$ be a Gaussian variable with mean zero and standard ...
2
votes
1answer
30 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
1
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1answer
35 views

Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx $$ However on one ...
1
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1answer
37 views

Properties of conditional expectation

In the probability book of Bauer it is claimed that for nonegative X and Y, or integrable X and Y we have $$ (1) \quad X=Y \, a.s. \Rightarrow \mathbb{E}(X \mid \mathcal{A}) = \mathbb{E}(Y \mid ...
1
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2answers
32 views

An $L^1$ convergence problem

Is the following true? If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$? ...
1
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2answers
53 views

let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous.

i'm trying to understand a proof of the following statement: let $X$ be a standard Gaussian random variable. Show that $(X,X)$ is not absolutely continuous. I'll write down the proof in such a ...
0
votes
1answer
47 views

Questions about expectation of stochastic integrals

I am considering the following SDEs: $$dX_1=-\theta(X_1-a_1)dt+\sqrt{X_1}(1-X_1)dW_1-X_1\sqrt{X_2}dW_2$$ $$dX_2=-\theta(X_2-a_2)dt-X_2\sqrt{X_1}dW_1+\sqrt{X_2}(1-X_2)dW_2$$ Here $W_1$ and $W_2$are ...
1
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2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
1
vote
1answer
39 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
0
votes
0answers
69 views

Expected value of correlated stochastic integrals

I do not understand the following result: Suppose $dz_\chi$ and $ dz_\xi$ are correlated increments of standard Brownian motion with $dz_\chi dz_\xi=\rho dt$ you have the following expectation ...
33
votes
5answers
4k views

Free throw interview question

I recently had an interview question that posed the following... Suppose you are shooting free throws and each shot has a 60% chance of going in (there is no "learning" effect and "depreciation" ...
1
vote
2answers
59 views

Why $E(X\mid X^2+Y)=0$ for $(X,Y)$ standard normal?

Please help me explain why $E(X\mid X^2+Y)=0$ for $(X,Y)$ standard normal, i.e $(X, Y) \equiv (0, I_2)$.
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
2
votes
3answers
159 views

We throwing $m$ balls to $n$ cells…

We throwing $m$ balls to $n$ cells randomly... At each cell can be more then one ball, or (of course) it can still empty. What is the expectation of the empty cells? I'd like to get any help! Thank ...
1
vote
1answer
49 views

How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
0
votes
1answer
68 views

Does $E[XY|Z]=E[XY]$?

Assume $E[X|Z]=E[X]$. Assume $Y$ and $Z$ are independent. Does $E[XY|Z]=E[XY]$? Can you prove it? My intuition says $E[XY|Z]=E[XY]$ but expanding the expectations into integrals I couldn't prove it. ...
0
votes
1answer
25 views

Expectation of exponential of Brownian motion

I want to compute the following expectation: $\mathbb{E}[\int_0^\infty-e^{-\mu t+\sigma W_t}dt]$ where $W_t$ is a brownian motion, $\mu$ and $\sigma$ constant. I am already stuck at computing the ...
11
votes
5answers
401 views

The Price is Right optimal play

The following situation happened on the Price is Right and I was curious about the optimal response. The rules are: A contestant rolls a wheel with 5 cent increments from 5 - 100 (20 numbers total). A ...
1
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0answers
19 views

Given the data set is the Bayesian estimation the best solution for solving the expected value?

I am very new to this. I have several measurements that from which I need to estimate a truth value. Each of them comes with an estimated error. I know that the observation error are biased (I don't ...
0
votes
1answer
25 views

Expectation of excess demand

Suppose random variable D has C.D.F. F. D is demand and y is supply in this case. Now, excess demand (D-y), D>y is lost and excess inventory (y-D), y>D is wasted. I have to find Expectation of lost ...
3
votes
2answers
50 views

Expected number of parallel tosses, where each coin gets heads at least once, of N unfair coins

A common expectation question is to ask "What is the expected number of tosses to get heads with an unfair coin?" This problem can be solved using the recursive equation E = p*1+(1-p)*(E+1), resulting ...
0
votes
0answers
17 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
0
votes
1answer
47 views

Expected Sum of n numbers after m random opeartions on range of n numbers. [duplicate]

I have been given 25 objects numbered from 1 to 25 and a set S = {-2, -1, 0, 1, 2}. I have to choose a random number from the given set and add it to all objects ...
1
vote
1answer
66 views

Expected Sum of n numbers after m random opeartions on n numbers.

I have been given $25$ objects numbered from $1$ to $25$ and a set $S = \{-2, -1, 0, 1, 2\}$. I have to choose a random number from the given set and add it to all the objects. I have to do this $5$ ...
1
vote
0answers
24 views

generalized expression required

suppose i have a set $ {0,1,2.......x-1}$ Now I am generating an i length sequence using the numbers from above set...${a0,a1,....ai}$ where all $ai$$>=0 $ and $ai<=x-1$ Note numbers may ...
4
votes
2answers
53 views

Why are stochastic processes with decreasing expected value called supermartingales?

I am curious to know why a process which has decreasing expected value is called a supermartingale. From a beginners perspective it would seem reasonable to have the following picture: ...
-1
votes
2answers
39 views

Find the Variance of X [closed]

I'm trying to find the expected value of random variable X $f(x)=\begin{cases} x, \ \ \text{if} \ \ 0 < x <1 \\ 2-x, \ \ \text{if} \ \ 1 \leq x <2 \\ 0, \ \ \text{elsewhere} \end{cases}$
1
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0answers
32 views

Simple question about showing independence [on hold]

How does one show that it is possible for a random variable $Z$ to be independent of $A$ but also not independent of $X$ where $X=1\{A>B\}?$ Under what circumstances can this be true, ignoring the ...
4
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2answers
39 views

Conditioning on information about the moments of a random variable is trivial

Say we have some random variable, $X$. Is it always trivial to condition on information about the moments of $X$? For example, suppose we know that $\mathbb{E}(X)$ is positive. But ...
1
vote
2answers
51 views

What is the expected value of max(x-y,0)?

For a normally distributed $X$ (with mean zero and variance $\sigma^2$) and a constant $Y$, I need to know $\mathbb{E}(\max(X-Y,0))$ I think that this should be $\mathbb{P}(X<Y)\times 0 + ...
1
vote
1answer
59 views

Show that convergence in the mean implies convergence of the means [closed]

Question: Let $X_n$, n = 1,... denote a sequence of real-valued random variables; $X_n$ is said to converge in mean if $\hspace{20mm}$$$\lim_{n\to\infty} E[|X_n-X|] = 0$$ Show that if $X_n$ ...
5
votes
1answer
81 views

What is the expected value of the number of randomly chosen real numbers between $0$ and $1$ needed to reach a sum of $1$? [duplicate]

My friend told me that the answer to this question was $e$, which intrigued me, but he refused to tell me why. My initial intuition was completely wrong. I thought that since the expected value of ...
0
votes
1answer
11 views

Further Conditioning upon already Conditional Expectation

Let's say that $Y$ as a sample space of $\{1,2\}$ and $Z$ has a sample space of $\{3,4\}$. I know that $E[X]=E[X|Y=1]P(Y=1)+E[X|Y=1]P(Y=2).$ Now suppose I now want to further condition upon $Z$. I ...
0
votes
0answers
29 views

A particular form of expectation

I am stucked in showing an equality concerning the expectation of a fonction $f \in \mathcal{L}_1(P) , ||f||_1 = \int_R |f(x)|dP(x)$ , where $P$ is a probability measure. The exercise is the ...
3
votes
1answer
33 views

$n$ players roll a die. For every pair rolling the same number, the group scores that number. Find the variance of the total score.

This is problem 3.3.3.(b) in Probability and Random Processes by Grimmett and Stirzaker. Here's my attempted solution: We introduce the random variables $\{X_{ij}\}$, denoting the scores of each ...
1
vote
2answers
87 views

Expectation of random variables ratio

Let $X_1, X_2, \dots, X_n$ be $n$ positive iid random variables. Then show that $$E\left(\frac{\sum_{j=1}^k X_j}{\sum_{i=1}^{n} X_i}\right) = \frac{k}{n}.$$ Because of the linearlity of the ...
3
votes
1answer
88 views

If $X_{i}$ are I.I.D and $n^{-1}\sum_{i=1}^{n}X_{i}$ converges a.s/in-distribution to a constant $a$ is it true that $a=\mathbb{E}[X_{i}] $?

The question itself is in the title. It is immediate by the strong law of large numbers that if $X_{i}$ had a finite first moment then we would have a.e convergence (and thus in probability and in ...
1
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0answers
20 views

Deriving joint distribution from expectation

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
1
vote
1answer
29 views

Estimate variance, how to find expected value of $x^2 [n]$

We have data $x_0, x_1, \ldots, x_{N-1}$ where the $x_n$'s are independent and identically distributed as ${\rm Normal}(0,\sigma^2)$. The estimate of $\sigma^2$ is $$\hat \sigma^2 = \frac{1}{N} ...
6
votes
1answer
52 views

Finding tight upper/lower bounds for $\mathbb{E}[\frac{1}{1+X^{2}}]$ where $X$ is a RV with $\mathbb{E}[X]=0$ and $\mbox{Var}(X)=\nu<\infty $

The question is pretty much in the title. My first thought was using Jensen's inquality to get some sort of lower bound. Since $\frac{1}{1+x^{2}}$ is convex on ...
0
votes
1answer
32 views

Strange identity in the proof of the Strong Law of Large Numbers

Above is an extract from the proof of the strong law large of large numbers with finite fourth moment. The $X_n$ are iidrv's with $\mathbb{E}(X_n)=\mu$ and $\mathbb{E}(X_n^4)<M$ for some ...
0
votes
3answers
72 views

Show that the E(|X|) is finite.

Show that if $E(X^2)<\infty$ then $E(|X|)<\infty$. My try: In other word, if $$\int x^2f(x)dx<\infty\Rightarrow\int xf(x)dx<\infty$$ for continuous case which $\int f(x)dx=1$ or $$\sum ...
2
votes
1answer
49 views

Expected value of log of 1+ a squared Gaussian random variable

If $X$ is standard normal, what is $$\mathbb{E}\log(1+X^2).$$ I see that $X^2\sim\mathrm{Gamma}(\frac 12,2),$ but is there a simple formula for the above (perhaps in terms of the polygamma ...
5
votes
2answers
59 views

Non-geometric way to calculate expected value of breaks?

In "50 Challenging Problems in Probability", question #43 is the following: "A bar is broken at random in two places. Find the average size of the smallest, of the middle-sized, and of the largest ...
0
votes
0answers
26 views

Variance under different probability measures

I am puzzled by the following problem: suppose $P \sim Q$ and $0 \leq E_{Q}(X) < E_{P}(X)$, where X is a nonnegative well-defined random variable. Is it possible to find the sign of $Var_{P}(X) - ...
0
votes
0answers
20 views

Simple question about conditional expectation

Let $X, Y,$ and $Z$ be random variables. Assume $X$ is independent of both $Y$ and $Z$. I know that $E[X|Z]=E[X]$. But is it true that $E[XY|Z]=E[X]*E[Y|Z]$? Sorry for not using formatting.