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Oct
26
comment Proving that $\sum_{a=1}^{b} \frac{a \cdot a! \cdot \binom{b}{a}}{b^a} = b$
Perhaps you could expand $\binom{b}{a}$ and cancel it with $a!$, grouping the resulting terms, but not sure what to do with it:$$ a \binom{b}{a} \frac{a!}{b^a} = a \frac{b!}{a! (b-a)!} \frac{a!}{b^a} = a \frac{(b-1)!}{(b-a)!b^{a-1}} = a \prod_{k=b-a+1}^{b-1} k $$
Oct
26
comment Why is the convergence value from an iteration the correct answer in this example?
You have to assume some conditions on the equations involved, in which case your iterative procedure would indeed converge. A simpler example of a similar technique would be Newtons' method for systems, e.g. here: google.com/…
Oct
26
comment How can you show that $E(Y\mid E(Y\mid X)) = E(Y\mid X)$?
Isn't $\mathbb{E}$ defined as an integral, how can you do this without integration?
Oct
26
comment I.I.D. random variables almost sure convergence
Just thinking out loud: if $X_i$ are iid, so are $\ln(X_i)$, and so $$\frac{1}{n} \sum_{i=0}^n \ln (X_i) \to \mathbb{E}[X_i] = 0$$ a.s. by KSSLN.
Oct
20
comment Characteristic functions of Poisson and normal distribution
@BozoVulicevic for (3),. just exchange the order of deriv and integral: $$\frac{d}{dt} \left[ \int f(t,x) dx \right] = \int \frac{d}{dx} \left[ f(t,x) \right] dx$$ under some nice conditions
Oct
20
comment Social welfare convergence in large assignment problems with random utilities
(2) does look very intuitive. Not sure about (1)
Oct
20
comment Characteristic functions of Poisson and normal distribution
@BozoVulicevic Taylor series: $$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$
Oct
20
comment Characteristic functions of Poisson and normal distribution
@BozoVulicevic ;-) thanks
Oct
20
comment How does one solve the following differential equation?
is pressure $P_1, P_2$ variable or constant?
Oct
20
comment fast computation of traiangular matrix with certain pattern
Do you want $A \vec{x}$ or $\vec{x} A$?
Oct
20
comment Finding fixed points via the Jacobian matrix eigenvalues
@stevetronix yes
Oct
20
comment Finding fixed points via the Jacobian matrix eigenvalues
@stevetronix please see the last couple of lines in the update
Oct
20
comment Stuck in a constant coefficient case question in linear algebra
@crysispeed I don't understand. Plug in the initial condition $y(-2)=1$ and also differentiate and plug in the second initial condition $y'(-2)=3$
Oct
20
comment If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines ax+by+c=0 is concurrent at which point?
please see my answer
Oct
19
comment What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?
another useful one, with quick convergence, is $$\pi^2 = 6\sum_{k=1}^n \frac{1}{k^2}$$
Oct
19
comment What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?
en.wikipedia.org/wiki/…
Oct
19
comment how to calculate compound interest when year is not whole?
@aura so what happens when you plug them into the formula? (Use $r = 0.1$) You will get the entire investment return. Subtract the principal to get just the earned interest
Oct
19
comment If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines ax+by+c=0 is concurrent at which point?
$$4a^2+9b^2+12ab-c^2 = (2a)^2 + (3b)^2 + 2 \cdot (2a) \cdot (3b) -c^2 = (2a+3b)^2 - c^2$$
Oct
19
comment how to calculate compound interest when year is not whole?
@aura so what are $P=?,r=?,n=?,t=?$ from your formula, what are the actual values?
Oct
14
comment Prove or Disprove: $m^2-n^2=2$ where m and n are integers. (Checking)
looks good to me