Timeline for Monotonicity of $\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c \right]$ in $n$ for $c>\mathbb{E}(X).$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2019 at 18:02 | comment | added | zhoraster | For a fixed $n$, consider such distribution: $P(X_1 = n) = 1-P(X_1=0) < 1/n$. Then $EX_1<1$ and for $k=1,\dots,n$ the probability $P(S_k >k) = P(\exists i\le k: X_i = n)$ is obviously increasing in $k$. | |
| Mar 16, 2019 at 11:47 | comment | added | kodlu | Interesting. Could you elaborate. | |
| Mar 16, 2019 at 11:16 | comment | added | zhoraster | Of course, it is not necessary: one can construct many examples of distributions which satisfy this property, but don't have a non-decreasing density (or even have no density at all). But some assumption is certainly needed. Actually, for each $n$ it is possible to construct a distribution such that $P(S_n > cn) > P(S_{n-1}>c(n-1)) >\dots >P(X_1>c)$ for some $c>EX_1$. | |
| Mar 14, 2019 at 10:23 | history | asked | kodlu | CC BY-SA 4.0 |