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Jun
25
comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)?
Thank you: we appreciate your respect for not double-posting on SE sites.
Jun
25
comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)?
(Migrating to Mathematics by request of the OP.)
Jun
11
comment What are the odds that the pattern, win lose lose, will happen 23 times in a row (69 rounds)?
This problem actually is much easier than any of the related ones. Your questions are answered simply by applying the definition of independence (which is what you must assume to answer them without further assumptions): the probabilities of independent events multiply. So go ahead and do the multiplications. You will notice that although the patterns will determine the probabilities, they do not change the fact that in every case you just multiply them all, so there really is nothing here involving combinatorial issues.
May
19
comment Proof for Mean of Geometric Distribution
Your reference gives three distinct derivations. The other two seem to answer your question.
Apr
28
comment Empirical likelihood method to Find the order of a parameter given a set of constraints
What is "$\lambda$"?? You provide no definition. BTW, a better way to get quick, insightful, creative help for questions like this is to provide motivation and background: explain what the important symbols $L$, $J_k$, and $S_k^r$ mean and represent. As it stands, these are just a bunch of abstract equations that would require exceptional effort from most readers (who are otherwise qualified to help you) to understand, which is going to limit (rather severely) your chances of receiving a good response, even after you tell us about $\lambda$.
Mar
7
comment Bivariate normal distribution: showing that linear combinations of joint Gaussians are Gaussian
This is merely equation (21) with $\rho$ used in place of that ratio of covariances.
Feb
18
comment Prove that $a_i\leq 0$ for $i=1,2,…,N-1$?
You might have the wrong sign in the equation you obtained: check it against the sequence $(0,-2,0)$ for $N=2$, for instance. As a hint, relax the conditions and suppose only that $a_{i+1}-2a_i+a_{i-1}\ge 0$ for $i=1,2,\ldots,N-1$. Can you still draw the desired conclusion?
Feb
11
comment Difference between Real Analysis and Probability Theory?
Nice little glitch in the SE technology: I have been able to upvote this answer twice--once on CV and once again here :-).
Feb
11
comment Difference between Real Analysis and Probability Theory?
+1 This answer gets to the heart of the matter. Analysis and probability have different interests: although they use similar sets of tools, they ask different questions and pursue almost completely different avenues of investigation. The two disciplines will nevertheless remain closely intertwined because insights from one can lead to progress in the other, much as (say) investigations of general relativity and quantum mechanics in physics have inspired advances in low-dimensional topology and operator theory, respectively.
Jan
30
comment What's a proof that the angles of a triangle add up to 180°?
@JoeZ. Or $-2\pi$: the complex arithmetic keeps track of angle orientation, too.
Dec
29
comment Interesting and unexpected applications of $\pi$
This is David H's answer in disguise: the ARE is algebraically related to (a) $e^0=1$ and (b) $\int_\mathbb{R}e^{-x^2}dx=\Gamma(1/2)=\sqrt{\pi}$. The appearance of $\pi$ is due to the duplication formula for the Gamma function, $\Gamma(z)\Gamma(1-z)=\pi\csc(\pi z)$, exhibiting $\Gamma$ as a kind of "square root" of a trig function. This is because $\Gamma(z)=\Gamma(z+1)/z$ implies $\Gamma$ has (simple) poles at $0,-1,-2,\ldots$, whence $\Gamma(z)\Gamma(1-z)$ has poles at $\mathbb Z$, strongly suggesting periodicity--which is why $\pi$ should show up!
Dec
28
comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
@Thank you for noticing that, Thomas. That is easily fixed by replacing one of the $c$'s by $1$; I will make the change.
Dec
24
comment Probability that there is sub-sequence of exact length
There is a lot of literature on this subject. The asymptotic distribution is known. I don't believe exact formulas for the case of finite $N$ are available.
Dec
24
comment Urn problem with balls
@Brian Thank you for that observation. It's very much in the spirit of this answer, which is an attempt to find as simple a solution as possible.
Dec
24
comment The point of contact of between two circles and common tangent at this point.
+1 for the simplified elementary approach. But note that the solution is incomplete: it has not (yet) addressed whether the circles are internally or externally tangent.
Dec
24
comment Urn problem with balls
It might be even simpler just to count the ways of choosing one ball in one urn and the other in the other: $(n+1)^2$. Divided by $\binom{2n+2}{2}$, this gives the answer directly.
Dec
24
comment Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $
@Brian You're right; there are many inductions lurking here. The one you point to comes down to asserting that the integers are closed under addition and multiplication. I'll accept that as immediately obvious. The crux of the matter, though, is that scalars commute with matrices, whence $(2\mathbb P)^n = 2^n \mathbb P^n$: that shows where the powers of $2$ come from.
Dec
15
comment Show that $\lim_{n\rightarrow\infty}{\displaystyle\sum_{i=1}^{n}{\frac{F_n}{2^n}}}=2$
The limit is $0$, not $2$. Use $(1 +\sqrt{5})/2$ instead of $2$ in the denominator to obtain a finite limit.
Dec
12
comment How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$
You ignore all terms except for the $k=1$ coefficient, which (upon multiplication by $1!=1$) yields the result you are looking for.
Dec
11
comment How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$
The result you reference makes it easily possible (via linear substitutions and completing the square) to compute the integral of $\Phi((w-a)/b)f(w;\mu,\sigma^2) \exp(\lambda (w-a)/b)dw$. The MacLaurin series of that function of $\lambda$ will yield integrals proportional to $\Phi((w-a)/b)((w-a)/b)^k f(w;\mu,\sigma^2)dw$ for $k=0, 1, 2, \ldots$; you want the case $k=1$.