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22 votes
Accepted

Expected value of the square root of a lognormal variable

The square root of a lognormal is itself lognormal. Indeed $Y^p$ is lognormal more generally. Note that if $X\sim N(\mu,\sigma^2)$ then $Y=\exp(X)$ is lognormal with the same parameters and vice versa....
Glen_b's user avatar
  • 285k
16 votes

Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose

To confirm a claim is not true, you don't "prove it". Instead, just provide a counterexample would be sufficient. You are actually on the right track. Any random vector $z$ with positive ...
Zhanxiong's user avatar
  • 20.4k
15 votes

Expected value of the square root of a lognormal variable

If Wikipedia is trustworthy enough for you, it says that $$ E(X^n) = \text{exp}\big(n\mu+\frac{1}{2}n^2\sigma^2\big) $$ for real or complex $n$, so you can use this formula with $n=\frac{1}{2}$. ...
Stephan Kolassa's user avatar
10 votes
Accepted

Is a product that has 4.9 stars from ten customers better than one that has 4.5 stars from a hundred customers?

Clearly, several online platforms like Trip Advisor etc. have implemented a sorting system that trades these two things off. If they all did it the same way I'd guess one might consider that a ...
9 votes
Accepted

Expectation of a function of the sample covariance matrix

This is a quite challenging and interesting problem as it calls for many classical results of multivariate Gaussian distribution and technical matrix operations. To begin with, note that since $\hat{\...
Zhanxiong's user avatar
  • 20.4k
9 votes
Accepted

Expected absolute deviation greater than standard Laplace

Consider a ReflectedGamma($b$, $c$) distribution (also known as a double Gamma distribution) with pdf $f(x)$: $$f(x) = \frac{ |x|^{c-1} \space e^{-|x|/b}}{ 2 \space \Gamma (c) \space \space b^c} \quad ...
wolfies's user avatar
  • 7,823
9 votes
Accepted

Expected value after $K$ Bernoulli trials where the $i$-th probability of success depends on the current number of successes

At the outset, note that you are specifying $1+2+3+\cdots + K = O(K^2)$ parameters for this distribution, so the very best solution will require $O(K^2)$ calculations. It will also require $O(K)$ ...
whuber's user avatar
  • 327k
9 votes

$E[(X+Y)^{a}] > E[(X)^{a}]$?

For all $x>0$, $y>0$ and $a>0$ you have $x+y>x$ so that: $$(x+y)^a>x^a.$$ Consequently, for any joint distribution of $(X,Y)$ where the support is on the strictly positive values for ...
Ben's user avatar
  • 127k
8 votes
Accepted

Expected value and variance of median

Caveat: The calculation below relies on the (not explicitly stated) condition that the sample $\{Y_1, \ldots, Y_n\}$ is drawn independently from $U(0, \lambda)$ once $\Lambda = \lambda$ was observed. ...
Zhanxiong's user avatar
  • 20.4k
8 votes

Expected value of decreasing function of random variable versus expected value of random variable

For the reason I stated in my comment, I assume what you are really interested is to find an example such that $E[g(X_1)] > E[g(X_2)]$ for some decreasing function $g$, and random variables $X_1$, $...
Zhanxiong's user avatar
  • 20.4k
7 votes

Interview Question: What is the probability they will be home in more than 30 minutes?

The student can only make it home in under 30 minutes by going straight there. The first coin flip effectively must come up heads, since the student will just stay at U re-flipping as many times as ...
Nuclear Hoagie's user avatar
7 votes
Accepted

expectation value, distribution function and the central limit theorem

You don't use CLT to get this result. What is needed is a direct evaluation of the term $E[S_n^3]$. To begin with, note that for $n \geq 3$: \begin{align*} S_n^3 = (X_1 + X_2 + \cdots + X_n)^3 = \...
Zhanxiong's user avatar
  • 20.4k
6 votes

How is that possible that simple arithmetic mean works well even for strongly skewed distribution?

What is your definition about "good"? I assume you want to say the bias of sample mean is zero, it is consistent, if so, in 1947, Hsu and Robbins proved that the arithmetic mean converges ...
Tuobang Li's user avatar
6 votes
Accepted

What is the fourth moment of a Euclidean Norm?

\begin{align} E[X^4] &= E[X^2X^2] \\ &= \text{Cov}(X^2,X^2) + E[X^2]^2 \\ &= \text{Var}(X^2) + E[X^2]^2 \end{align} Note that $X^2 = \| M^T p \|_2^2 = p^T M M^T p$, which follows a ...
mhdadk's user avatar
  • 5,100
6 votes
Accepted

Inequalities involving expectations

Note that all the $Y=1$ conditions can be dropped, as everything is conditioned on $Y=1$, and $Y$ appears nowhere except in the conditioning part of the statements. Replacement condition: (2'). $E(W|...
jbowman's user avatar
  • 40k
6 votes

Expected value of largest eigen value of sample correlation matrix

It depends on the correlation between the features (dimensions), i.e., the data. Firstly, there will be some zero eigenvalues because when $p \gg n$, $\mathbf{R}$ will be positive semi-definite ($\...
wjktrs's user avatar
  • 860
6 votes

Expected value of decreasing function of random variable versus expected value of random variable

A counterexample: $\mathcal{X} = \{0,1,3\}$, $f_1(x) = \{0.5,0,0.5\}$, $f_2(x) = \{0,1,0\}$, and $g(x) = \{5,2,1\}$. $$\mathbb{E}_1[X] = 1.5 > \mathbb{E}_2[X] = 1$$ $$\mathbb{E}_1[g(X)] = 3 > \...
jbowman's user avatar
  • 40k
5 votes
Accepted

Monte-Carlo integration with importance-sampling

Importance sampling means using a substitute density $q(\cdot)$ when integrating an arbitrary integrable function $H(\cdot)$ $$\int H(y)\,\text dy$$ since $$\int H(y)\,\text dy=\int \frac{H(y)}{q(y)}q(...
Xi'an's user avatar
  • 106k
5 votes
Accepted

Given $r\gt 0$, how to get $\mu_r = E[|U|^r]$ where $U\sim N(0,1)$?

It is just a matter of integration: \begin{align*} & E[|U|^r] = \int_{-\infty}^\infty |u|^r\frac{1}{\sqrt{2\pi}}e^{-u^2/2}du \\ =& 2\int_0^\infty u^r\frac{1}{\sqrt{2\pi}}e^{-u^2/2}du \\ =& ...
Zhanxiong's user avatar
  • 20.4k
5 votes

Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose

Here's a discrete counter-example, if that's easier to wrap your head around. Let $$ \begin{align} P(z_0=1 \wedge z_1=1) &= 0.5 \\ P(z_0=-1 \wedge z_1=-1) &= 0.5 \end{align} $$ Then $$ E[z] = ...
Passer By's user avatar
  • 151
5 votes
Accepted

Expectation of product of sample averages

Even in very advanced mathematics it helps to study simple examples. Part of the art of reading and learning mathematics is to construct such examples for yourself. This answer illustrates the ...
whuber's user avatar
  • 327k
5 votes

What conditions are there on the exponent $p$ such that $\underset{\mu}{\arg\min}\left\{\mathbb E\left\vert X-\mu\right\vert^p\right\} $ must exist?

The choice $p=0$ is the only one such that the arg min in question always exists. For any $p>0$ we can find a distribution $F$ such that $\mathbb{E} \vert X - \mu \vert^p$ does not exist for any $\...
picky_porpoise's user avatar
5 votes
Accepted

Probability algorithm on strings

Ok, so you have strings $Z_n$ that are exactly half 0s and half 1s and you have $X_n$ that have exactly one more 1. One way to generate these sorts of strings is to generate a $Z_n$ uniformly at ...
Thomas Lumley's user avatar
5 votes

Expected Value Chi Square distribution

Your (5) "I changed $\frac{(n−1)s^2}{\sigma^2}∼\chi^2_{n-1}$ to $\frac{(n−1)s^2}{\chi^2_{n-1}}∼\sigma^2$" does not make much sense as $\sigma^2$ does not have a distribution while your $(n-...
Henry's user avatar
  • 40.3k
4 votes

Expectation of the softmax transform for Gaussian multivariate variables

I'd like to complement @myscience's answer in terms of an alternative approach mentioned in that article. Basically, the approach expands the function (in your case, the softmax transform) to 2nd-...
Kevin's user avatar
  • 81
4 votes

The expectation of the inverse of a negative binomial random variable?

In this paper (equation (3), derived in Appendix A) you can find a closed-form expression for that: $$ \mathbb E \left[\frac 1 x \right] = \sum_{i=1}^{n-1} \frac{(-1)^{i+1}}{n-i} \left(\frac{p}{1-p}\...
Luis Mendo's user avatar
  • 1,099
4 votes

Provide an intuitive example of the linearity of expectation

Sticking to your example, here's an intuitive way to understand the linearity of expectation (at least to me): Consider the $i$th-consecutive pair in the sequence of 9 cards of two different colors, 4 ...
Tran Khanh's user avatar
4 votes

Certain approximation in the setting of three expectation values does not make sense to me

Indeed, it is incorrect to state that $$\mathbb{E}_{x^*} \left[\mathbb{E}_y\left[\left\{\mathbb{E}_{x}\left[f(x^*,y,x)\right]\right\}^n\right]\right]= e^{n\mathbb{E}_{x*}\left[\mathbb{E}_y\left[\log\...
Xi'an's user avatar
  • 106k
4 votes
Accepted

Expectation of Wordle score

Here's a simple parametric approach which works by treating the failures as right-censored data. It gives an answer of $4.1065$. We are going to begin by assuming that $X$, the number of incorrect ...
knrumsey's user avatar
  • 8,317
4 votes

Reducing Variance in Estimating the Exponential Average of Random Variables

This is effectively the same as asking: How can we estimate an average? The exponential is just a re-expression of the variable with different numbers. Examples where either $X$ or $e^X$ might be ...
Matt F.'s user avatar
  • 5,052

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