Mathematics

Questions tagged [expectation]

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

3719 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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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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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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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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59
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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 you…
svenkatr
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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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34
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6 answers

Proving $E[X^4]=3σ^4$

Given a random variable $X\sim\mathcal N(0,\sigma^2)$, how can we prove that $E[X^4]=3\sigma^4$? I am having trouble even starting with the proof.
foooyoh
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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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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 Jensen's…
Ramil
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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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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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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 Y]]=E[X]…
Boby
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