Tagged Questions
5
votes
4answers
364 views
Why does this expected value simplify as shown?
I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is:
...
7
votes
2answers
269 views
Probability that a sequence repeats itself
Given an infinite sequence $a_n$ of uniformly random integers $0$ to $9$, what is the probability there exist an integer $m$ such that the sequence $a_1$ to $a_m$ is equal to that from $a_{m+1}$ to ...
21
votes
6answers
963 views
Proofs of $\lim\limits_{n \to \infty} \left(H_n - 2^{-n} \sum\limits_{k=1}^n \binom{n}{k} H_k\right) = \log 2$
Let $H_n$ denote the $n$th harmonic number; i.e., $H_n = \sum\limits_{i=1}^n \frac{1}{i}$. I've got a couple of proofs of the following limiting expression, which I don't think is that well-known: ...
40
votes
5answers
6k views
Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
...
14
votes
1answer
215 views
Density of odd numbers in a sequence relating base 2 and base 3 expansion
Define the function
$$f(4n)=6n+1\\
f(4n+1)=6n+2\\
f(4n+2)=6n+3\\
f(4n+3)=6n+5$$
and the sequence $u_0=2$, $u_{k+1}=f(u_k)$.
Let $d_1\le d_2$ be the lower and upper asymptotic density of odd numbers ...
6
votes
1answer
221 views
Probabilistic proof of existence of an integer
The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$.
Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
2
votes
0answers
143 views
Find a tight upper bound of the following expectation.
I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
2
votes
1answer
81 views
Behaviour of Two Coupled Sequences Towards a Stable Distribution
The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
1
vote
3answers
36 views
Proof that $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1} = \frac{2^{m+1}-1}{m+1}$ [duplicate]
Recently I needed to compute $E[\frac{1}{X+1}]$ where $X\sim Bin(m, \frac 1 2)$.
While expanding, I came across the sum $\sum_{k=0}^m \binom{m}{k}\frac{1}{k+1}$, which I was unable to solve. Plugging ...
1
vote
2answers
246 views
What is linearity of Expectations?
In reading about the average case analysis of randomized quick sort I came across linearity of expectations of indicator random variable I know indicator random variable and expectation. What does ...
0
votes
1answer
51 views
Expected value of a certain sort of game
Please check my work, expected value calculations are often of the sort where you get an answer but there's no "check", so to speak.
Imagine you have a scenario with the following rules:
Inputs ...
