Intuition The one core ideaA continuous distribution is a way of smearing exactly one unit of "probability mass" over a range of numbers, like spreading one kilogram of butter over a slice of bread — thick where values are likely, thin where they are rare. Everything in the parent topic is just: choose the shape of the smear, then measure areas under it.
Before you can read a single formula in the parent note, you need to own the alphabet it speaks in. Below is every symbol and idea it uses, ordered so each one leans only on the ones above it. Nothing appears before it is earned.
Definition The number line and[ a , b ] number line is just the ruler of all real numbers stretching left (negative) and right (positive). An interval [ a , b ] a b
a left edge (smallest allowed value).b right edge (largest allowed value).Square bracket [ ] ( )
Picture it: a horizontal ruler with two vertical fence-posts at a b
Why the topic needs it: every distribution lives on some stretch of the line — Uniform on [ a , b ] [ 0 , ∞ ) [ 0 , 1 ] stage on which probability is spread.
Definition Random variableA random variable X X x
Picture it: a pointer that will drop somewhere onto the number line, but you don't yet know where. Different runs of the experiment drop it in different places.
Why the topic needs it: the whole subject is about describing where the pointer tends to land . X X
P ( something ) 0 1 0 1 0.5
P ( X ≤ x ) X x The symbol ≤ >
Intuition Why probability becomesarea for continuous things
For a continuous X exactly one precise number (say exactly 3.0000 … 0 x range ," and that chance is an area under a curve.
Why the topic needs it: this single move — probability = area — is the engine of the entire chapter. Every formula is secretly "compute this area."
Definition Function notationf ( x ) function f x f ( x ) f f ( x ) x
Definition Probability density function (PDF)The probability density function f ( x ) height tells you how concentrated the probability is near x f ( x ) ≥ 0 ( height never negative ) , ∫ − ∞ ∞ f ( x ) d x = 1 ( total area = 1 ) .
Common mistake "The heightf ( x ) Why it feels right: it sits between us and probability. Fix: height is density (probability per unit length ), not probability. It can even exceed 1 area — height ×
Why the topic needs it: f ( x ) f is choosing the distribution.
Definition The integral sign∫ a b f ( x ) d x ∫ a b f ( x ) d x "add up the area under the curve f x = a x = b Read the pieces:
∫ S um.d x x f ( x ) d x f ( x ) d x The ∫ a b a b
this tool and not plain multiplication?
If the curve were a flat line of constant height c c × ( b − a ) bend : height changes as you move. Multiplication needs a constant height; the integral is the only tool that correctly totals a varying height by chopping the region into slivers thin enough that each is flat. That is exactly the question a probability asks: "how much area under a curvy shape?"
Why the topic needs it: P ( a ≤ X ≤ b ) = ∫ a b f ( x ) d x
Definition Cumulative distribution function (CDF)F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t running total of area collected from far left up to x 0 1 t x
F ′ ( x ) derivative F ′ ( x ) slope (steepness) of the CDF curve at x f ( x ) = F ′ ( x ) slope of accumulated area; accumulated area is the sum of density.
Why the topic needs it: the parent derives every distribution by writing down either f or F ∫ ′
E [ X ] μ E [ X ] balance point of the density — where the curve would balance on a pivot. We call it the mean and name it μ μ = E [ X ] = ∫ − ∞ ∞ x f ( x ) d x . f ( x ) d x x
σ Var ( X ) = E [ X 2 ] − μ 2 average squared spread — how far, squared, values sit from the mean on average. Its square root σ standard deviation , a spread in the same units as X
squared distance?
Plain distance from the mean averages to 0
Why the topic needs it: every distribution is summarised by its ( μ , σ 2 )
e e − λ t e ≈ 2.718 e − λ t 1 t = 0 decays smoothly toward 0 t rate λ
Intuition Why exponentials appear in waiting problemse − λ t only shape with the property "the fraction lost per unit time is always the same." That constant-forgetting is exactly what a memoryless wait needs — see Poisson Process . So the tool is chosen because it matches the physics , not by accident.
Why the topic needs it: Exponential, Gamma, and Normal densities all contain an e
Definition Symbols you will meet
α , β shape parameters; they bend a curve's silhouette.Γ ( α ) Gamma Function , a smooth stand-in for the factorial: Γ ( n ) = ( n − 1 )! B ( α , β ) Beta Function , the exact area that normalises a Beta curve.Φ N ( 0 , 1 ) π ≈ 3.14159 2 π
Why the topic needs it: these are the normalising constants — the numbers you divide by so the total area equals exactly 1
Number line and interval a to b
CDF F and derivative F prime
Exponential decay e to minus lambda t
Greek and special functions
Cover the right side and see if you can answer before revealing.
What does the height f ( x ) not )? Density = probability per unit length; it is not a probability. Only area under f f ( x ) 1
Why do we ask P ( a ≤ X ≤ b ) P ( X = x ) X The chance of hitting one exact real number is 0
What single instruction does ∫ a b f ( x ) d x "Add up the area under the curve f x = a x = b
Why an integral rather than multiplication for area under f The height varies; multiplication needs constant height, so we chop into slivers thin enough to be flat and sum them.
How are the PDF f F F ( x ) = ∫ − ∞ x f ( t ) d t f ( x ) = F ′ ( x )
What is the mean μ The balance point of the density; μ = ∫ x f ( x ) d x
Why square the distance from the mean in variance? So overshoots and undershoots cannot cancel to zero, and larger misses count more.
What does e − λ t t λ Decays smoothly from 1 0 λ
What job do Γ B 2 π Φ They are normalising constants / area tools ensuring total probability is exactly 1 Φ
