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Oct
23
comment What is the expected number of flips that are needed?
I have given a solution here of the same problem with $5$ consecutive. In order to make the argument accessible to people not familiar with Markov chains, the solution did not mention them explicitly, though you will recognize transition probabilities.
Oct
23
reviewed Leave Open Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer
Oct
23
comment Counting permutations that respect a partial order
You are welcome. The case of $2$ that you quoted is covered by a shorter version of the same argument. And the idea works in the same way for arranging $q$ types of coins.
Oct
23
revised Counting permutations that respect a partial order
added 420 characters in body
Oct
23
answered Counting permutations that respect a partial order
Oct
23
comment Proof of a sum of positive divisors
@Lucian: It follows directly from the above identity and multiplicativity.
Oct
23
reviewed Reopen Inequality: Find Min $S=\frac{a}{\sqrt{1-a}}+\frac{b}{\sqrt{1-b}}$
Oct
23
reviewed Leave Open How many will not be selected in repeated tries?
Oct
23
answered Probability Puzzle: Exactly one of two specific balls among $N$ balls in $n$ draws.
Oct
23
comment Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.
We don't need the restriction, it just makes things confusing. Given any $P$, any $N$, there is a sequence of $N$ consecutives each divisible by at least $P$ primes. This was done in the solution referred to. I don't really want to write down the proof, it would be a near duplicate.
Oct
23
comment Integrating $x\cdot{\cosh(x^2)}$
For $x\cosh(x^2)$, use substitution. The function $\cosh(x^2)$ of the title (but not of the question) does not have an elementary antiderivative.
Oct
23
answered Question About Notation Nested Quantifiers.
Oct
23
comment Given a joint PDF verify that it is a joint density function
Anything other than $1$.
Oct
23
comment Area and integration question, is this area under the curve?
If it says the exact area between the $x$-axis and $\dots$ then I would say integrate $(x-1)(x-2)(x-3)$ from $1$ to $2$, add the integral of $-(x-1)(x-2)(x-3)$ from $2$ to $3$.
Oct
23
awarded  Enlightened
Oct
23
comment How to seperate out a variable from a log
I do not see the $\ln(1+\frac{r}{4})=\ln(\frac{5r}{4})$.
Oct
23
comment How to seperate out a variable from a log
We have $20\ln(1+r/4)=\ln((1+r/4)^{20})=4/3$. So $(1+r/4)^{20}=4/3$ and therefore $1+r/4=(4/3)^{1/20}$. (There are other ways.)
Oct
23
awarded  Nice Answer
Oct
23
revised $\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence
added 116 characters in body
Oct
23
answered $\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence