omega 18.0%
input
X = -2
output
Y = -2
rare low outcome
Probability Mechanics Lab
Build the core probability objects by hand: sample spaces, random-variable maps, expectation, variance, conditioning, and transformations.
probability mechanics
Expectation
A weighted center of mass. Probabilities are the weights; random-variable values are the positions.
E[Y]
1.04
Var(Y)
2.72
P(event)
32%
active atoms
8/8
Sample space atoms
Each card is one possible outcome with probability mass.
omega 212%
input
X = -1
output
Y = -1
low outcome
omega 310%
input
X = 0
output
Y = 0
neutral
omega 420%
input
X = 1
output
Y = 1
common middle
omega 518%
input
X = 1
output
Y = 1
same value, different atom
omega 612%
input
X = 2
output
Y = 2
upper outcome
omega 710%
input
X = 3
output
Y = 3
high outcome
omega 810%
input
X = 4
output
Y = 4
tail outcome
Output distribution
Mass that lands on the same output value is added together.
Y = -28.0%
Y = -112%
Y = 010%
Y = 138%
Y = 212%
Y = 310%
Y = 410%
Current transform: .
Transform
diagnosis
Expectation is balance, not the most likely atom
The mean can sit between values that never occur. It is the point where weighted pull from the left and right balances.
why this matters
This is the language behind ML data
Losses, predictions, labels, gradients, and test metrics are random variables. Expectation is average behavior, variance is instability, conditioning is slicing the data-generating world, and transformations are how models create new quantities to measure.