# Questions tagged [expected-value]

Questions about the expected value of a random variable.

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### Problem in finding expectation of two consecutive faces in consecutive throws of a die

I came up with a question while doing a project on simulation on some experiments regarding dice. The question was: On average, how many times must a 6-sided die be rolled until two rolls in a row ...
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### Integrating Conditional Probabilities

I have a function : F(a,b,c) I want to take the Expected Value of this function in the following form: E[ F(a,b,c) | a, c) I know that for a single variable, the Expected Value is E(x) = integral(x*p(...
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### What is $\mathbb{E} \bigg\lvert \int f(x)\Pi(dx)\bigg\rvert^2$ for a random measure $\Pi$?

As mentioned in this wiki page we know that $$\mathbb{E}\int f(x)\Pi(dx)=\int f(x)\mathbb{E}\Pi(dx)=\int f(x)\nu(dx),$$ where $f$ is any measurable function and $\Pi$ is a random measure with ...
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### Bound on the probability of the size of a random walk [closed]

Let $S_n=X_1+\dots+X_n$, where $X_1,X_2,\dots$ are independent with $\mathbb{E}(X_m)=0$ and $\mathbb{E}(X_m^2)=\sigma_m^2\in(0,\infty)$ for all $m\geq1$. Let $\mathcal{F}_n=\sigma(X_1,\dots,X_n)$. I ...
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### coefﬁcient of kurtosis in higher moments of random variables

I don't quite understand why the $-3$ from the $\gamma_2$ equatioon is not used in the answer for the question below. It's even mentioned in the answer too! 3.6.21. Calculate the coefﬁcient of ...
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### Question about derivative of an Expected Value

Let $Z_0(t) \approx V_0e^{\lambda_0 t}$ be a branching process where $V_0 \sim \text{Exp}(\lambda_0/a_0)$, $\lambda_0 = a_0-b_0$ and $a_0,b_0 \in \mathbb{R+}$. I do not understand one step from this ...
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### Question on Conditional Expectation. with sum of Bernouli random variables.

X is a uniform random variable on $[n]=\{1,2,\dots ,n\}$ and for every $i\in[n]$ We define$Y_i \sim Ber(\frac{1}{2})$ ,such that all The variables are independent(all The Y's and X). now we have to ...
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### Why are these equations describing the variance of residuals equivalent (single-factor model in finance)? [closed]

Residual Equations The above equations describe the residual variance in the single-factor model in finance. I'm struggling to understand why they are equivalent, specifically why the fourth and fifth ...
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### Inequality involving moments of a distribution [closed]

Let X be a real random variable. Under what conditions on the distribution do we have that $$\mathbb{E}( X^{2n + 2}) \geq \mathbb{E}( X^{2n}) \mathbb{E}( X^{2})$$ for all integer $n$? I tried using ...
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### Probability mass function and expected value in lottery

I can buy one lottery ticket out of two available. In the first lottery I can win \$100 with probability 0.1, and the price of ticket is 10. In the second lottery, I can win \$50 with probability 0.1 ...
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### Taking expectation in proof of Markov inequality

In Probability and Computing (Mitzenmacher & Upfal), Markov inequality is proven roughly as follows: (Theorem) Markov's Inequality. Let $X$ be a random variable that assumes only nonnegative ...
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### On the expectation of conditional expectation [duplicate]

Let $X,Y,Z$ be real, integrable, continuous random variables. It is known that $E[E[X|Y,Z]]=E[X]$. What happens if we take the "expectation" on only one of the conditioning random variables? ...
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### Expected Value of Slot Machine

I have a probability question that has to do with slot machines. Here is how the game works: There are two reels, one on the left and one on the right. There are several symbols on each reel, one of ...
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### How to find the expected value of an exponential curve?

I have a function of the form $$y = \left(\frac{a}{b}\right)^{1/T} - 1$$ where $T$ is a future point in time, such that $$0 < T \leq \infty.$$ Question 1: Given constant values for $a$ and $b$, ex:...
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### Expected time until k failures among n items each independent and having exponential distribution

This is the question I am trying to solve. "Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution ...
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### Question about conditional expectation fact

This fact comes from *Concentration of Measure for the Analysis of Randomised Algorithms * by Dubhashi and Panconesi (page 76, equation 5.2). Let $X$ and $Y$ be two discrete random variables and two ...
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### Surprisingly simple expected time for the "range" of a Brownian motion to extend beyond $a$ - is there a martingale method?

I will call the range $R(t)$ of a standard Brownian motion the difference between its maximum $M(t) = \max_{0\leq s \leq t} B(t)$ and its minimum $m(t) = \min_{0\leq s \leq t} B(t)$. That is, I am ...
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### Show that $\mathbb{E}\Big[\mathrm{ln}\big(\frac{X}{\mathbb{E}[X]}\big)\Big] < 0$

We know that $X > 0$ and that $\mathbb{E}[X] < \infty$. Show that $\mathbb{E}\Big[\mathrm{ln}\big(\frac{X}{\mathbb{E}[X]}\big)\Big] < 0$. Could someone show me a way to prove it? Thanks.
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### Expected value as well as variance of observing a TH string in coin flips

I'm working with a problem where I need to calculate the expected value and the variance of the amount of coin flips needed until we observe a a "TH" - string in a coin flip experiment. In ...
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### Flipping $n$ coins until they're all heads
Suppose you have $n$ (fair) coins: You flip them until they're all heads in the following sense: Suppose you flip all of them first and get $k$ heads. This is round $1$ You remove these heads from the ...