Null hypothesis (H0): Octopus Paul doesn’t have the ability to predict. Or, the probability that he predicts correctly on each event is 1/2.
Data: In 2010 World Cup, Octopus Paul correctly predicted the outcomes of 8 games out of 8 games.
p-Value: (the probability to obtain the data assuming the null hypothesis is true): 1/2^8 = 0.0039
Statistical significance threshold: alpha = 0.05
Conclusion: as pvalue < alpha, we conclude that the null hypothesis should be rejected. Loosely speaking, octopus Paul does have prediction power.
What is the limit of this infinite exponential?
Solution 1:
$$x=\sqrt{2}^x$$
This leads to x=2 or 4
Solution 2:
$$x=[\sqrt{2}^\sqrt{2}]^x$$
This leads to x not equal to 2 or 4
Solution 3: MatLab simulation of series
$$\sqrt{2}, \sqrt{2}^\sqrt{2}, [\sqrt{2}^\sqrt{2}]^\sqrt{2},…$$
leads to infinity
Solution 4: MatLab simulation of series
$$\sqrt{2}, \sqrt{2}^\sqrt{2}, \sqrt{2}^{[\sqrt{2}^\sqrt{2}]},…$$
leads to 2
Then what’s is value of this tower of sqrt(2)?
Number of people visiting this blog fluctuates periodically with period 1 week. Basically on weekends few people visit.
Frequency analysis
Basel problem: find the exact value of
$$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+… = ?$$
Solution:
We have by Taylor expansion:
$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+…$$
Dividing through x we have,
$$\frac{\sin(x)}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+…$$
Note, the root of \(\sin(x)/x\) is at \(n\pi\), so we can rewrite \(\sin(x)/x\) as
$$\sin(x)/x=(1-x/\pi)(1+x/\pi)(1-x/{2\pi})(1+x/{2\pi})…$$
If we multiply out the product and collect the \(x^2\) terms, we see the \(x^2\) coefficient is
$$-\frac{1}{\pi^2}(1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+…)$$
But from the original expansion of \(\sin(x)/x\), the coefficient of \(x^2\) is -1/6. These two coefficient must be equal; thus
$$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+… = \frac{\pi^2}{6}$$
What is the value of
$$i^i$$
\(i\) being the imagery unit.
This beautiful number indeed equals to
$$i^i=e^{-\pi/2}=0.2079…$$
The name of “temporal lobe” follows “temporal bone”. But why “temporal”? The reason is that when people get old, the first region in the head which grows white hair is this region (see picture below). Thus this region is kind of a “time” region. That’s where “temporal” comes from. (I get this from some textbook but I forget which one.)
