# Questions tagged [expected-value]

The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.

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### Expectation of a random walk that can't go below zero

Suppose we have a random walk $S_n$ that is constrained to be positive or zero, that is: $$S_0 > 0$$ $$S_{i+1} = \max(S_i+x_i,\space 0)$$ $$x_i \sim N[\mu,\sigma^2]$$ Can we analytically ...
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### Finding the Cauchy Principal Value Mean of the pdf for $Z=XY$ where $\mathbb{E}[Y]$ does not exist

Let's first define the Cauchy Principal Value Mean (PVM). For a continuous random variable $V$ with pdf $f_{V}(v)$, the PVM of $f_{V}$ is \mathrm{PV}(\mathbb{E}[V])=\lim_{a\to\infty}\...
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### Expectation of a strictly increasing function

Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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### Conditional expectations by conditioning on functions of random variables

I have conjectured the following: Let $f:\mathbb{R}\supseteq A \rightarrow B \subseteq \mathbb{R}$ be an injective function. Let $X$ be a random variable with support $A$ and $Y$ be some random ...
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### Is the expected value a valid decision-making method in a very short term?

This might be related to game theory more than statistics, but I decided to ask this question here. Let's assume you're offered a lottery. There are a hundred balls in a bowl: 99 white balls and one ...
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### Calculating the variance of a weight-average when the weights also have a variance

Assume there is a series of random variables $X_1$, $X_2$, ..., $X_N$ representing a series of values to be weight-averaged, and a corresponding series of random variables $W_1$, $W_2$, ..., $W_N$ ...
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### How do I solve $E\left[ E \left(X|Z \right) E\left( Y|Z \right)\right]$?

I am trying to solve $E\left[ E \left( \mathbf{X}|\mathbf{Z} \right) E \left( \mathbf{Y}|\mathbf{Z} \right) \right]$, (where $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ are random variables) but I am ...
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### Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} \mathbb{E}\left[\frac{f(X)}{f(Y)...
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### An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
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### Smooth expectations outside the exponential family

At page 85-86 of Young and Smith "Essentials of Statistical Inference" there is an interesting result. If $X$ is a r.v. distributed according to the exponential family and $\phi(x)$ is a bounded (but ...
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### A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$
I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...