[Verse 1]
When probability dances with numbers in space
Expectation finds the balance place
Sum up each value times its chance to be
That's your center, probability's key
For discrete cases, sigma does the sum
For continuous curves, integrals come
[Chorus]
E of X is where the masses meet
Variance shows how far they scatter and retreat
E of X squared minus E of X all squared
That's the formula that keeps calculations paired
Linear always for expectation's call
Even when variables depend through it all
[Verse 2]
Variance measures how the data roams
How far each point wanders from its home
Take deviations, square them one by one
Average those squares when calculation's done
But there's a shortcut that's much more neat
Expected squares minus expected's repeat
[Chorus]
E of X is where the masses meet
Variance shows how far they scatter and retreat
E of X squared minus E of X all squared
That's the formula that keeps calculations paired
Linear always for expectation's call
Even when variables depend through it all
[Bridge]
Scale by A, shift by B, here's the transformation rule
Expected values: A times E plus B, that's the basic tool
But variance only cares about the stretch
A squared times variance, shifts don't affect the fetch
[Verse 3]
When combining random quantities
Expectations add with perfect ease
Independent or tangled up tight
Expected sums always sum up right
But variance needs more careful thought
Add covariance to what you've got
[Chorus]
E of X is where the masses meet
Variance shows how far they scatter and retreat
E of X squared minus E of X all squared
That's the formula that keeps calculations paired
Linear always for expectation's call
Even when variables depend through it all
[Outro]
Center and spread, the fundamental pair
Expectation's linear everywhere
Variance tells you how the chaos flows
Together they reveal what randomness knows