$$ f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}, \quad x \geq 0, \quad \sigma >0 $$ |

$$ f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}, \quad x \geq 0, \quad \sigma >0 $$ |

My childhood since the times of Kindergarten was full of simple scientific experiments. Here are some which I could remember:

- Calculated the number of photo-receptors of my retina by using trigonometry after measuring my sight sharpness at a known distance. (13 yo)
- Obtained a formula for angle between two planes if the line of their intersection, and a line connecting two non-intersection points of the planes are given. (16 yo)
- Obtained and programmed a formula to obtain the distance between two given locations on Earth, assuming the Earth surface is spheric. (14 yo)
- Discovered a formula to calculate the sum 1+2+3+...+n if n is given. (12 yo)
- Soldered a solar panel from a set of solar cells, and used it for audio cassette player. (15 yo)
- Made an automatic switch to control my air humidifier by a dry branch of fir that ascends or descends depending on the level of air humidity.
- Disassembled a video projector from 1950s. (5 yo)
- Read books, such as "ABC of Astronomy", "Science and Universe". (3 yo)
- Planted cucumbers, water-melons and other plants, while trying to experiment what happens if I add various chemicals to the soil.
- Collected chemical elements and their compounds. (14 yo)
- Tried to extract Hypericin from St John's wort using chromatography in UV light.
- Extracted oxygen from heated KMnO4.
- Fused metals: Copper, Lead, Tin, Aluminum, etc.
- Soldered...
- Played with Electrolysis. Lacking batteries for a cassette player, tried to make them from electrolytes and electrodes.
- Tried to extract N2O from NH4NO3 (but fearing its carcinogenic properties, didn't proceed).

(15 yo) - Bought and used a rotary evaporator for water distillation at home. (14 yo)
- Tried to obtain ciano-compound by making a discharge between carbon electrodes (taken from batteries) in nitrogen atmosphere.
- Made Iodine azide (an explosive compound) by putting crystals of Iodine into water solution of NH3. (14 yo)
- Tried to capture Iodine from its ethanol solution by putting copper wires into the solution.
- Tried to mix Aluminum powder, Magnesium slivers, KMnO4 and Sulfur powder, and ignite it.
- Tried to make my fingerprint photo by dipping my fingers into aluminum powder, then sticking it to a transparent sticky sellotape, and sticking it to photo-film.
- Tried to make art from chemistry. Examples: http://inyuki.deviantart.com/gallery/#/d2b14w , http://inyuki.deviantart.com/#/d2bxem , http://inyuki.deviantart.com/gallery/#/d2bxbu .
- Made a solution of Luminol, which glows in the presence of complex Iron ions. (15 yo)
- I tried to make a mirror by depositing silver from Silver nitrate onto glass. (14 yo)
- I read a book "Numbers and Their Science" about natural numbers, number systems, prime numbers, Goldbach's conjecture. (12 yo)
- I tried to make a concentrated NaCl solution, and tried to make a freezer by cooling the solution to -20 C, and circulating it through copper tubes...
- When ill, I enjoyed reading books about Immunology and Genetics.
- I tried to make stage smoke by heating glycerol on a pancake, and directing an air current onto its surface.
- I tried to make St John's wort extract using izopropil alkohol.
- I calculated the Centrifugal acceleration, and tried to make a centrifuge from an electric motor, then measured how many G`s can a cockroach withstand.
- I made a terrarium, intended to keep a snake, but kept a little lizard, a turtle. I read books about herpetology.
- I tried to inflate balloons with Hydrogen by putting Aluminum coins into water solution of NaOH.
- I tried to mix glycerol with nitric acid in order to obtain nitroglycerin. (Very carefully)
- I tried to make Hydrogen sulphide by putting ZnO into H2SO4.
- I learned multiplication table of octal number system, and used it instead of decimal for making my calculations. (13 yo)
...

Many other.