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

# Questions tagged [expectation]

3755 questions

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### I roll a die repeatedly until I get 6, and then count the number of 3s I got. What's my expected number of 3s?

Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable $X$. What is $E[X]$?
Thoughts: I expect to roll the die 6 times before 6 appears (this part is…

nettle

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### Would you ever stop rolling the die?

You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect square, then you lose, and you go home with…

Newb

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### Expected number of unpecked chicks - NYT article

In this article, the winner of the math competition answered this question correctly:
In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of…

AAC

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### Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?

Choose a random number between $0$ and $1$ and record its value. Do this again and add the second number to the first number. Keep doing this until the sum of the numbers exceeds $1$. What's the expected value of the number of random numbers needed…

user25329

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### Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space
$ (\Omega, \mathscr{A}, \mu ) $ with $\mu(\Omega) = 1 $, and a random variable $ X :…

smiley06

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### If a 1 meter rope is cut at two uniformly randomly chosen points, what is the average length of the smallest piece?

If a $1$ meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece?
I got this question as a mathematical puzzle from a friend. It looks similar to MathOverflow question If…

svenkatr

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### what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the average value of the distribution. Did I get it…

user2340452

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### On average, how many times must I roll a dice until I get a $6$?

On average, how many times must I roll a dice until I get a $6$?
I got this question from a book called Fifty Challenging Problems in Probability.
The answer is $6$, and I understand the solution the book has given me. However, I want to know why…

Sidd Singal

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### How does a disease spread through a triangular network?

Consider a population of nodes arranged in a triangular configuration as shown in the figure below, where each level $k$ has $k$ nodes. Each node, except the ones in the last level, is a parent node to two child nodes. Each node in levels $2$ and…

MGA

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### A disease spreading through a triangular population

I have run into this problem in my research, which I'm presenting under a different guise to avoid going into unnecessary background.
Consider a population that is connected in a triangular manner, as shown in the figure below. The root node has a…

MGA

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### Knight returning to corner on chessboard -- average number of steps

Context: My friend gave me a problem at breakfast some time ago. It is supposed to have an easy, trick-involving solution. I can't figure it out.
Problem: Let there be a knight (horse) at a particular corner (0,0) on a 8x8 chessboard. The knight…

SSF

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### Prove that $\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$ for i.i.d $X$ and $Y$

Let $X$ and $Y$ be two independent identically distributed random
variables with finite expectation $\Bbb{E}(X) = \Bbb{E}(Y) < \infty$. Prove that
$$\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$$
I think that this inequality may follow somehow from…

Ramil

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### Number of moves necessary to solve Rubik's cube by pure chance

Suppose, random moves are made to solve Rubik's cube. A move consists of
a $90$-degree-rotation of some side. The starting position is also random.
What is $E(X)$, where $X$ is the number of moves until the cube is solved ?
How many moves must be…

Peter

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### Upper and Lower Bounds on $Var(Var(X\mid Y))$

Are there any particular properties that
\begin{align*}
Var(Var(X\mid Y))
\end{align*}
satisfies so that we can derive any upper and lower bounds on it.
For example, if we replace $Var$ with expectation we have
\begin{align*}
E[E[X\mid…

Boby

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### supremum of expectation $\le$ expectation of supremum?

Suppose that $X$ is an arbitrary random variable, is the following is true for any function $f$:
$$\underset{y\in \mathcal Y} \sup \mathbb E\big[f(X,y)\big] \le \mathbb E\big[\underset{y\in \mathcal Y} \sup f(X,y)\big]?$$
If $f$ is convex in $X$,…

syeh_106

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