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

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Probability & Expected Value Questions

I am working on some question. I have the solutions to these questions so I am not looking for the answer I really need help with how I should approach these question, being guided through them. ...
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Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
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Distance of random span to a vector

I've been batteling with the following problem: Assume we have a diagonal matrix $D \in \mathbb{R}^{l \times l}$, a vector $\beta \in \mathbb{R}^l$. Next we simulate a random matrix (Idea inspired by ...
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Expected Value with graph theory

A group of $n\geq 3$ people is sitting at a round table, so that each person has two neighbors,one clockwise neighbour and one counter clockwise neighbour. Each person flips a fair and independent ...
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Expected Value for coin flips for five heads.

Consider a coin that comes up heads with probability $p$ and tails with probability $1-p$ We flip this coin (independently) and stop as soon as it comes up heads for the 5th time. Let X be the random ...
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Normal distribution squared probability

Let $X_1,X_2,X_3,X_4$ be independent standard normal random variables and $Y=X^2_1+X^2_2+X^2_3+X^2_4$. Find the probability that $Y≤3$. Enter your answer as a decimal and make sure that at least $10$ ...
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23 views

Expected Value of Truncated Random Variable

I have what appears to be a relatively simple question, but am struggling to understand how to go about answering it. The general question is as follows: What is the expected value of $S_{I}$, ...
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Expectation of modulus of normal distribution.

I consider random variable $\xi \in N(o, \sigma^2) $. How to find teh expectation: $\mathbb{E}(|\xi|)$? It seems to be connected with the variance of $\xi$, but in which way?
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Easy mean-variance problem

Find the mean and variance of X, where X is the number of distinct results of a 12-sided die rolled 5 times (e.g. {3,11,12,3,11} returns X=3).
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Contitional Expectation of Sum of two uniform rv [on hold]

(apologies for posting a more specific version of an earlier question). Let $X$ be uniformely distributed on $[0,1]$ and $Y$ be uniformely distributed on $[m-k,m+k]$ with m being a number in $[0,1]$ ...
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26 views

Decomposition of sum of two independent random variables

Let $X$ and $Y$ be two indpenent r.v. How can I get an expression for: $$E[X|X+Y=a]$$ where $a$ is a constant? In other words, is there a general rule to recover the expected value of $X$ when all I ...
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“Reverse” distribution-tails

Chernoff, Markov and Chebyhev all give some upper bound for tail probabilities, e.g. Chebyshev gives us $Pr[|X-E[X]| \geq t] \leq \frac{Var[X]}{t^2}$. This is quite helpful, but what if I would ...
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25 views

Mean and variance: Gaussian is the most conservative assumption

"given only the mean and variance of a distribution, the most conservative assumption that can be made about the distribution is that it is a Gaussian having the given mean and variance" I've read ...
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bound of | E[X/Y] - E[X]/E[Y] |

Is there some bound for $ | \mathbb{E}[X/Y] - \mathbb{E}[X]/\mathbb{E}[Y] | $ ? where $X$ and $Y$ are both summation of a fixed number of Bernoulli random variables and a constant that is >0, which is ...
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35 views

Expectation and variance of seats in a movie theater

A movie theater has a row of 10 seats. Each seat is occupied with probability $1/2$ independently of the other seats. A visitor is "comfortable" if he or she is not at the edge of the row and both ...
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50 views

Show that $E[(X-E[X])^2] = E[X^2]-E^2[X]$

I'm trying to use linearity of expectation, but I seem to be failing somewhere. So if we expand the squared expression: $$(X-E[X])^2 = (X-E[X])(X-E[X]) = X^2 - 2XE[X] + E^2[X]$$ So we have: $$E[X^2 ...
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21 views

calculating variance of a random variable

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other three ...
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Expectation and variance of the number of couples sitting at the same table

There are $2n$ people at a party, making $n$ couples. These $2n$ people are seated randomly at $n$ tables that have $2$ seats each. Let $X$ be the number of couples that sit at the same table. Find ...
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Meaning of the expectation of a conditional variance

I'm reading page 2 of conditional probability. The author states that if $\text{Var(X|Y)}$ is treated as a random variable then the expectation is $\text{E[Var(X|Y)] = E[E[X^2|Y]] - E[E(X|Y)]^2}$ My ...
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26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
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Expected value of multiplied and squared Wiener Process

May someone help me how to calculate the following thing: E0[z^2[2] Exp[-2 z[2]] ] Where z[2] is Wiener process. How to find exact expected value? I am new to this stuff, not sure how to do this. ...
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30 views

Expected value for linear scan before encountering one of k elements

You are given a random permutation of $n$ distinct elements labeled $[0..n-1]$. If you go through them one at a time what's the expected number of items you'll have to go through before you encounter ...
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Approximating $\pi$ using expected values [closed]

We approximate $\pi$ by choosing random points in a square and seeing what fraction fall in the inscribed circle. If we choose $k$ random points, the expectation $E[X]$ of the number $X$ of points in ...
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Matrix of Expectation of Random variables Update [closed]

I am not a math guy, but here I have encounter a problem about finding an inverse matrix, which the original matrix are elements of expectation of random variables. I think it is an optimization ...
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32 views

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$

$\rho_\gamma(X)=\frac{1}{\gamma} \log \mathbb{E}[e^{-\gamma X}]$ is a convex risk measure, but it fails the subadditivity property in order to be called coherent. A mapping ...
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21 views

Bounding the quotient of random variables

I have two non-negative random variables $X, Y$ with finite expected values and variances, and I want to bound $E(X/Y)$ from above. I was reading these notes and they do a two-variable version of ...
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22 views

Product of two random variables together with conditional density

Let $X_1$ and $X_2$ be two real valued random variables such that we have the conditional density of $X_1$ given $X_2$, i.e. $$\mathbb P(X_1\in M\mid X_2) = \int_M \phi(x_1\mid X_2)dx_1$$Also, let $h$ ...
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Let $X_1,X_2,…,X_n$ ~$Uniform(0,1)$ and $Y=max[X_1,X_2,…,X_n]$, find $E(Y)$!

I don't really have any good starting point. I guess Y is a function of the X's and so I might be able to apply the theorem about the expectation of a function, but I don't know how to deal with this ...
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Expectation of 3 exponential distributed random variables

Suppose a component independently produced by 3 machines. The lifetime of the components made by machine 1, 2, and 3 are exponentially distributed with parameters A, B, and C respectively. Machine ...
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Lowerbound on expectation of log sum of Bernoulli random variables?

I'm looking for a lower-bound on $E\left[\log \left(B + \sum_i a_i X_i\right)\right]$ where $X_i$ are Bernoulli random variables with $p(X_i = 1) = q_i$ and $a_i > 0, B > 0$. Because $X_i$ is ...
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Number of storms in a rainy season

This is a follow-up to my previous question. Now instead of finding a probability I would like to now find the expectation too. I will restate the question and my solution below. I would appreciate if ...
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How to get from this probability formula to the one I need?

I'm working on a gambler's ruin problem where a player starts out with $i$ money, and 'winning' is when their total money reaches $N$ (ie they will keep playing until they reach N or run out of money, ...
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Expectation of $\mathbf{v}\mathbf{v}^\top$ when $\mathbf{v}$ is an independent Bernoulli vector.

It has been a while since I took probability class, so I dare to ask. Suppose I have $\mathbf v$, a column vector of length $n$. Each entry is independently chosen from $\{+1, -1\}$ so that for each ...
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Gaussian independent, Mean, expectation, variance

Let $X$ and $Y$ be two independent Gaussian variables with zero mean and variance $\sigma^2$. Define:$$Z = |X-Y|.$$(a) Show that $\operatorname{E}[Z]= 2 \sigma / \sqrt{\pi}$. (b) Show that ...
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Inequality: $\mathbb E[(X-\mathbb E[X\mid \mathcal H])^2] \le \mathbb E[(X-Z)^2]$

Let $X$ be a random variable on the probability space $(\Omega, \mathcal A, P)$. Suppose $X$ is square-integrable and $\mathcal H$ is a sub $\sigma$-algebra of $\mathcal A$. Then $\mathbb E[X\mid ...
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Expected value of 2 Poisson distributions

Let $X$ and $Y$ be independet Poisson random variables with parameters $\lambda$ and $\mu$. I have to calculate $E((X+Y)^2)$ . What I did: $E[(X+Y)^2]=E[X^2]+E[Y^2]+2EXEY$ I know that ...
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How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
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Gradient of expectation of e times its conjugate

Assuming e is a complex series of data, why equation below holds for it's gradient? $$ \bigtriangledown E\{e(n) e^*(n)\}= 2E\{ \bigtriangledown(e(n))e(n)^* \}$$ Where $\bigtriangledown $ is the ...
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1answer
25 views

Expected Value of lowest-x-in-sample for [0,1] distribution

Let's assume take N random numbers in the [0,1] interval. What is the expected value of the lowest number in the sample? Running a few simulations it really seems ...
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26 views

Am I doing this Expected Value question properly?

P(Success) = (1/1000) 1-P = (999/1000) Say Mark wins +699 Dollars if Success else Mark wins -1 Dollars now say he does this 200 times how much money will he have? $$\sum\limits_{i=1}^{200} ...
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60 views

Draws from the uniform distribution are taken until the sum exceeds 1. What is the expected value of the final draw?

I was thinking about this question after a related problem: what's the expected number of draws for the sum to exceed 1? For that problem, the answer is known and is a surprising result ...
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Coupon collector problem

Let $T$ be the time to collect all $n$ coupons, and let $t_{i}$ be the time to collect the i-th coupon after $i − 1$ coupons have been collected. Think of $T$ and $t_{i}$ as random variables. ...
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52 views

How should I define E(X|Y) if I want to show that E[E[X|Y,Z]|Z]=E[X|Z]?

I'm taking an introductory probability class and recently had a homework problem asking whether ${\bf E}[{ \bf E}[X|Y,Z]|Z]={\bf E}[X|Z]$ is true (it is). I'm trying to understand the proof for this ...
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Extension to Classical Coupon Collectors Problem

If there's n different coupons. Instead of ordering coupons one-by-one until you collect all n coupons as in the traditional 'Coupon Collector Problem', what if the coupons came in packs of m coupons. ...
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Finite expectation of renewal process

Let $T_n$ be a random variable with $T_n=X_1+...+X_n$ where the $X_i$'s are iid. Further we set $N(t)=max\{ n: T_n\leq n\}$ with the property $\Pr(N(t)<\infty)=1$. I want to prove that ...
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applying iterated expectation when conditioning on multiple random variables

The law of iterated expectations tells us that ${\bf E}\big [{\bf E}[X\, |\, Y]\big ]={\bf E}[X]$. Suppose that we want apply this law in a conditional universe, given another random variable $Z$, in ...
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Determining expected value with Linearity of Expectation

A store sells n flavors of chips. When you place an order the store sends you 1 bag of chips, chosen uniformly at random from the n different flavors, independently of previous orders. Tiffany would ...
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233 views

Expected number of triangles in a random graph of size $n$

Consider the set $V = \{1,2,\ldots,n\}$ and let $p$ be a real number with $0<p<1$. We construct a graph $G=(V,E)$ with vertex set $V$, whose edge set $E$ is determined by the following random ...
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finding the unspecified ${\bf E}[X]$ and $\rm var(X)$ given the expectation of higher powers of $X$

Homework Problem: It is known that a for a standard normal random variable $X$, we have ${\bf E}[X^3]=0$, ${\bf E}[X^4]=3$, ${\bf E}[X^5]=0$, ${\bf E}[X^6]=15$. Find the correlation coefficient ...