# Questions tagged [expected-value]

Questions about the expected value of a random variable.

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### Take a random walk on Pascal's triangle, without revisits: Does the final number have infinite expectation?

Let's take a random walk on Pascal's triangle, starting at the top. Each number is in a regular hexagon. At each step, we can move to any adjacent hexagon with equal probability, but we cannot revisit ...
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### Expected number of strikes to kill a $3$-headed dragon

You want to slay a dragon with $3$ heads. There is $0.7$ chance of destroying a head and $0.3$ chance of missing. If you miss, a new head will grow. $X$ is a random variable for the number of rounds ...
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### Expected winning score in a round robin tournament

Suppose you have a tournament consisting of 100 players in which each player plays each other player exactly once. In any given match, the chance of either playing winning is equally likely, and there ...
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### Finding expected value with replacement [closed]

I have balls numbered 1 to 100 in a bag. I take one out, record it, and put it back into the bag. I repeat this 1000 times. My questions are: What is the expected value E[X] of never picking a number?...
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### Covariance of the product of two random variables with another random variable

Let $X\sim \mathrm{Bin}(p_x,1)$ , $Y\sim \mathrm{Bin}(p_x,1)$ and $V\sim \mathrm{Bin}(p_v,1)$ be three binomial random variables, that are NOT independent from each other. Note that $X$ and $Y$ are ...
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### When does $k$-th central moment relate to $k$-th power of expectation?

Given a bounded random variable $X$. When exactly is the $k$-th central moment $E (X-\mu)^k$ upper bounded by the $k$-th power of the expectation $\mu^k$? This seems to be the case for bounded, ...
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I have an $100$-sided die. Every round, I roll the die, and can either choose to receive the amount shown by the die, or pay $1 to reroll the die and continue. If I want to find the expected value of ... 6 votes 2 answers 175 views ### Conjecture: If$x_k$are random in$(0,\pi/2)$then expectation of$\frac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$is$(\pi/2)^{2n-6}$for$n>2$. Let$E(n)=\text{expectation of }\dfrac{\prod_{k=1}^n\tan x_k}{\sum_{k=1}^n\tan x_k}$where$x_k$are independent uniformly random real numbers in$\left(0,\frac{\pi}{2}\right)$. Is the following ... 3 votes 2 answers 69 views ### Probability and average height of two lines intersecting above the x-axis For context, I know at least basic calculus and what random variables are, but not much about doing calculations with random variables. I came up with a problem for myself where there are two points ... 4 votes 2 answers 304 views ### Card game Puzzle You have a special deck with 11 cards, with 10 red cards and 1 joker. starting with$1 we play a game: shuffle the deck we can choose to draw the top card. if it's red, we double our money, but if it'...
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Is my thinking correct? Let's suppose there are two independent random complex vectors, $a,b\in\mathbb{C}^N$. Additionally, both vectors have a zero mean, and their covariance matrices are defined (e....
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### $\int_0^{\pi/2}\int_0^{\pi/2}\frac{(\tan\alpha)(\tan\beta)}{\tan\alpha+\tan\beta} d\alpha d\beta=(0.9999999913...)(\pi/2)$? Seriously?

In the diagram, $\alpha$ and $\beta$ are independent uniformly random real numbers in $\left(0,\frac{\pi}{2}\right)$. What is $\mathbb{E}(h)$? Superimposing a cartesian coordinate system, the ...
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### Product of 2 dice game

You are playing a game where you a paid the product of 2 dice a) How much would you pay to have the opportunity to reroll any 1 of the dice? b) How much would you pay to roll 3 dice and discard the ...
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### Determining bet sizes given odds [duplicate]

Crossposted on Quant SE Recently, I was asked the following question in an interview with a prop trading firm. You are given the opportunity to make money by betting a total of 100 bucks on the ...
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### Does $E^2(X)/E(Y) \le E(X^2/Y)$ hold

Let $f(x),g(x)$ be $\mu$ measureable functions with $g(x) > 0$ almost surely. Does $$\frac{\left(\int f(x)\mu(x)dx\right)^2}{\int g(x)\mu(x)dx} \leq \int \frac{(f(x))^2}{g(x)}\mu(x)dx$$ hold? I ...
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### Probability of guessing correct number between 1-100

You need to guess a number between 1-100 and after each guess I'm going to tell you if you guessed too high or too low. If you guess it on the first try you get \$5, on the second try \$4, then \$3, \$...
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This question was asked in an interview of a Quantitative finance company. You have $\textit{n}$ fair coins with either heads or tails facing up arranged in a row. You play a game in rounds described ...