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Probability for sum of $n$ uniform $(0, 1)$ being larger than sum of $n+1$ uniform $(0,1)$

Let $X = \sum\limits_{i=1}^{n} \left(1-U_i\right) +\sum\limits_{i=1}^{n+1} W_i = n-\sum\limits_{i=1}^{n}U_i + \sum\limits_{i=1}^{n+1} W_i$. It is clear that $X \sim Bin\left(2n+1, \frac12\right)$ and ...
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  • 5,549
1 vote

Distribution of N smallest uniformly distributed random variables

Average distant doesn't matter that much: of you by luck got one element very close to $a$, this doesn't mean you can relax requirements on next. Note that $K$ consists of points closest to $a$ iff ...
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  • 8,901
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Difference between two difference uniform distributions

Note: Always keep an eye on the supports. First, we let Z=X−Y. Since the distributions are independent, we can multiply their densities: No, not quite. The joint pdf between $Z$ and $Y$ is such a ...
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1 vote

Difference between two difference uniform distributions

Since both of $X$ and $Y$ are uniformly distributed and they are independent. The joint pdf is $f_{XY}=\frac{1}{(b-a)(d-c)}$. The total $2D$ region is the rectangle region with vertices coordinates $A(...
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  • 5,259
1 vote

When does a ud mod $1$ sequence remain ud mod $1$ if infinitely many terms are deleted or changed?

You can remove any subsequence of density $0$. Obviously it may fail if you remove a subsequence of density $> 0$.
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