# Tag Info

## Hot answers tagged expected-value

### Find expected value using CDF

The result extends to the $k$th moment of $X$ as well. Here is a graphical representation:
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### Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

This is another illustration of Jensen's inequality $$\mathbb E[\log X] < \log \mathbb E[X]$$ (since the function $x\mapsto \log(x)$ is strictly concave] and of the more general (anti-)property ...
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### Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?

Consider two values symmetrically placed around $0.5$ - like $0.4$ and $0.6$ or $0.25$ and $0.75$. Their logs are not symmetric around $\log(0.5)$. $\log(0.5-\epsilon)$ is further from $\log(0.5)$ ...
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### Brain-teaser: What is the expected length of an iid sequence that is monotonically increasing when drawn from a uniform [0,1] distribution?

Here are some general hints on solving this question: You have a sequence of continuous IID random variables which means they are exchangeable. What does this imply about the probability of getting a ...
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### I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

I would like to offer a very simple, intuitive explanation. It amounts to looking at a picture: the rest of this post explains the picture and draws conclusions from it. Here is what it comes down to: ...
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### Why do we care more about test error than expected test error in Machine Learning?

Why do we care more about $\operatorname{Err}_{\mathcal{T}}$ than Err? I can only guess, but I think it is a reasonable guess. The former concerns the error for the training set we have right now. ...
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### What is the expected value of the logarithm of Gamma distribution?

This one (maybe surprisingly) can be done with easy elementary operations (employing Richard Feynman's favorite trick of differentiating under the integral sign with respect to a parameter). We are ...
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### Why is the expected value named so?

Imagine that you are in Paris in 1654 and you and your friend are observing a gambling game based on sequential rolling of a six sided dice. Now, gambling is highly illegal and busts by the gendarme ...
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### Sample two numbers from 1 to 10; maximize the expected product

If you don't get to the smart covariance trick by Glen B, then you could also consider the following approach which is one level of abstraction lower Step 1: consider computing the hard way by adding ...
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### Let X,Y be 2 r.v. with infinite expectations, are there possibilities where min(X,Y) have finite expectation?

Let's find a general solution for independent variables $X$ and $Y$ having CDFs $F_X$ and $F_Y,$ respectively. This will give us useful clues into what's going on, without the distraction of ...
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### What is the expected value of the logarithm of Gamma distribution?

The answer by @whuber is quite nice; I will essentially restate his answer in a more general form which connects (in my opinion) better with statistical theory, and which makes clear the power of the ...
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### What is a random variable and what isn't in regression models

This post is an honest response to a common problem in the textbook presentation of regression, namely, the issue of what is random or fixed. Regression textbooks typically blithely state that the $X$ ...
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### Expectation of a function of a random variable from CDF

When $F$ is the CDF of a random variable $X$ and $g$ is a (measurable) function, the expectation of $g(X)$ can be found as a Riemann-Stieltjes integral \mathbb{E}(g(X)) = \int_{-\infty}^\infty g(x) ...
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### Constructing example showing $\mathbb{E}(X^{-1})=(\mathbb{E}(X))^{-1}$
Let's construct all possible examples of random variables $X$ for which $E[X]E[1/X]=1$. Then, among them, we may follow some heuristics to obtain the simplest possible example. These heuristics ...