I.
Distributions
f(x) = (1 / σ√2π) · e−(x−μ)² / 2σ²
01 / HISTOGRAM
N(μ, σ²)
bins24
02 / DENSITY
μ ≈ 0.00
drift±0.5
03 / CUMULATIVE
F(x) = ∫−∞x f(t) dt
F(0)0.500
04 / Q-Q PLOT
Φ−1(p)
n160
05 / BOX PLOT
Q1, Q2, Q3
IQR1.34
06 / GENERATOR
x ~ N(0,1)
rate10/s
II.
Random Walks
Xt+1 = Xt + εt, ε ~ N(0,1)
07 / WALK 1D
ΣXt
step0
08 / WALK 2D
(x,y) ∈ ℤ²
paths3
09 / BROWNIAN
dX = σ dW
σ0.07
10 / STEP HIST
P(±1) = 0.5
balance0.50
11 / DISPLACEMENT
|Xn| ~ √n
rms0.00
12 / GALLERY
ensemble
walks9
III.
Convergence
limn→∞ (1/n) Σ Xi = E[X]
13 / COIN CONV.
p̂n → 0.5
p̂0.500
14 / DICE AVG.
x̄n → 3.5
x̄3.50
15 / SAMPLE MEAN
x̄ ~ N(μ, σ²/n)
n30
16 / CI WIDTH
CI = x̄ ± z·σ/√n
95%±0.40
17 / MONTE π
π ≈ 4·Nin / N
π̂3.000
18 / CLT
x̄ → N(μ, σ²/n)
samples0
I.
Distributions
Probability Laboratory