# Tag Info

Accepted

• 20.8k
Accepted

### 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 ...
• 129k
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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: ...
• 328k
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• 328k
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### Is the median preserved for any strictly monotonic mapping?

The conjecture is true and your disproof is flawed. The flaw in your steps occurs when you make a change-of-variables in the initial step but do not change the range of integration accordingly. ...
• 129k
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• 328k
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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 ...
• 328k

• 2,586

### Why is $x + x = 2x$, but $X + X \neq 2X$?

[An earlier version of the question asked for an answer that completely avoided mathematics; this answer was an attempt to give some intuitive motivation, at a similar level to the document being ...
• 286k
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### What is the difference between random variable and random sample?

A random variable, $X:\Omega \rightarrow \mathbb R$, is a function from the sample space to the real line. This is a deterministic formula that can be as simple as writing down the number a die lands ...
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### If X and Y are uncorrelated, are X^2 and Y also uncorrelated?

Even if $\operatorname{Corr}(X,Y)=0$, not only is it possible that $X^2$ and $Y$ are correlated, but they may even be perfectly correlated, with $\operatorname{Corr}(X^2,Y)=1$: ...
• 23.7k
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### When can I not replace a random variable with its mean?

If you replace a missing value by some point estimate, you disregard all its variability. Thus, you will not propagate all the original variability to your model. Your parameter estimates will appear ...
• 128k
$E[(X-E[X])^2] =0 \implies X = E[X]$ Thus $X$ is almost surely constant. A better description for such random variables is that it follows a degenerate distribution.
### For i.i.d. random varianbles $X$, $Y$, can $X-Y$ be uniform [0,1]?
No. If $Y$ is ever (with positive probability) $> X$, then $X - Y < 0$, so it can't be $U[0,1]$. If $X$ and $Y$ are iid, $Y$ can not be guaranteed (i.e., with probability $1$) to not be $> X$...