Math notes

From raju

dummy

useful links

paradoxes

This applies when profiling code. If you have a hotspot that takes 99% of the time and optimize it until your whole program is twice as fast, now that hotspot takes 98% of the time.

partial fractions

Horner's method

Newton's identities

likelihood function

The definition of the likelihood function for continuous case:

[math] \mathcal{L}(\theta | x) = p_{\theta}(x) [/math]

The left term means “the likelihood of the parameters [math]\theta[/math], given data [math]x[/math]”.

Here the notation [math]\mathcal{L}(\cdot | \cdot)[/math] does not mean the same as the conditional notation. Another representation of the likelihood that is often used is [math]\mathcal{L}(x; \theta)[/math], which is better in the sense that it makes it clear that it’s not a conditional, however, it makes it look like the likelihood is a function of the data and not of the parameters.

Ref:- http://blog.christianperone.com/2019/01/a-sane-introduction-to-maximum-likelihood-estimation-mle-and-maximum-a-posteriori-map/

averages

average of squares > square of averages

ex:- for [3,4]
average of squares = (3**2 + 4**2) / 2 = 12.5
square of averages = ((3 + 4)/2)**2 = 3.5**2 = 12.25

simplifications I came across

[math] \frac{x}{(1-\alpha x)(1-\beta x)} = \frac{1}{\alpha-\beta}\left[ \frac{1}{1-\alpha x} - \frac{1}{1-\beta x}\right] [/math]

laplace transform

wallis product

mental math

convert from miles to km

Double four times and divide by ten as soon as possible (i.e. the first time you see a zero on the end, drop it). This amounts to multiplying it by 1.6. Actual formula is 1 mi = 1.60934 km.

Ex:- 30 mph -> 3 (drop the zero) -> 6 -> 12 -> 24 -> 48 kmph.

No need to count how many times you are doubling. Look for something between 1x and 2x of the original number.

Ex:- 135 mph -> 270 -> 27 -> 54 -> 108 (keep going, less than 135) -> 216 kmph (stop here, less than 2* 135 = 270)

pounds to kilos

Divide by 2, less 10%. This amounts to multiplying it by 0.45. Actual formula is 1 lb = 0.453592 kg

Ex:- 160 lbs = 80 - 8 = 72 kg

kilos to pounds

Multiple by 2, add 10%. This amounts to multiplying it by 2.2. Actual formula is 1 kg = 2.20462 lbs