Motivating Benfords law by rotating a circle Motivating Benfords law - PowerPoint PPT Presentation

Motivating Benford’s law by rotating a circle Motivating Benford’s law by rotating a circle 1 / 6

Consider a set of naturally-occurring data Motivating Benford’s law by rotating a circle 2 / 6

Consider a set of naturally-occurring data—river lengths, for instance: Motivating Benford’s law by rotating a circle 2 / 6

Consider a set of naturally-occurring data—river lengths, for instance: Benford’s law is the probability distribution that describes the first-digit frequency of a data set like this. Motivating Benford’s law by rotating a circle 2 / 6

Consider a set of naturally-occurring data—river lengths, for instance: Benford’s law is the probability distribution that describes the first-digit frequency of a data set like this. � ℓ + 1 � p ( ℓ ) = log 10 , ℓ ∈ { 1 , . . . , 9 } . ℓ Motivating Benford’s law by rotating a circle 2 / 6

In this presentation, we will focus on the powers of 2: { 2 k : k ∈ N 0 } . Motivating Benford’s law by rotating a circle 3 / 6

In this presentation, we will focus on the powers of 2: { 2 k : k ∈ N 0 } . We will show that Benford’s law governs their first-digit frequency. Motivating Benford’s law by rotating a circle 3 / 6

In this presentation, we will focus on the powers of 2: { 2 k : k ∈ N 0 } . We will show that Benford’s law governs their first-digit frequency. The heart of the argument relies on a theorem, and we need some definitions to understand it... Motivating Benford’s law by rotating a circle 3 / 6

✶✷✳ Pr♦❜❛❜✐❧✐t② Pr❡s❡r✈✐♥❣ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ▼✉❝❤ ♦❢ ♦✉r ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ❤❛s r❡✈♦❧✈❡❞ ❛r♦✉♥❞ t❤❡ ❧♦♥❣ t❡r♠ ❜❡❤❛✈✐♦r ♦❢ s❡q✉❡♥❝❡s ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❲❡ ❝♦♥t✐♥✉❡ t❤✐s t❤❡♠❡ ✇✐t❤ ❛ ❝✉rs♦r② ❧♦♦❦ ❛t ❡r❣♦❞✐❝ t❤❡♦r②✳ ❘♦✉❣❤❧② s♣❡❛❦✐♥❣✱ ❡r❣♦❞✐❝ t❤❡♦r❡♠s ❛ss❡rt t❤❛t ✉♥❞❡r ❝❡rt❛✐♥ st❛❜✐❧✐t② ❛♥❞ ✐rr❡❞✉❝✐❜✐❧✐t② ❝♦♥❞✐t✐♦♥s t✐♠❡ ❛✈❡r❛❣❡s ❝♦♥✈❡r❣❡ t♦ s♣❛❝❡ ❛✈❡r❛❣❡s✳ ❆s ✉s✉❛❧✱ ✇❡ ❜❡❣✐♥ ✇✐t❤ s♦♠❡ ❞❡✜♥✐t✐♦♥s✳ ❉❡✜♥✐t✐♦♥✳ ❆ s❡q✉❡♥❝❡ X 0 , X 1 , ... ✐s s❛✐❞ t♦ ❜❡ st❛t✐♦♥❛r② ✐❢ ( X 0 , X 1 , ... ) = d ( X k , X k +1 , ... ) ❢♦r ❛❧❧ k ∈ N ✳ ❊q✉✐✈❛❧❡♥t❧②✱ X 0 , X 1 , ... ✐s st❛t✐♦♥❛r② ✐❢ ❢♦r ❡✈❡r② n, k ∈ N 0 ✱ ✇❡ ❤❛✈❡ ( X 0 , ..., X n ) = d ( X k , ..., X n + k ) . ❲❡ ❤❛✈❡ ❛❧r❡❛❞② s❡❡♥ s❡✈❡r❛❧ ❡①❛♠♣❧❡s ♦❢ st❛t✐♦♥❛r② s❡q✉❡♥❝❡s✳ ❋♦r ✐♥st❛♥❝❡✱ ✐✳✐✳❞✳ s❡q✉❡♥❝❡s ❛r❡ st❛t✐♦♥❛r②✱ ❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ s♦ ❛r❡ ❡①❝❤❛♥❣❡❛❜❧❡ s❡q✉❡♥❝❡s✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❛ st❛t✐♦♥❛r② s❡q✉❡♥❝❡ ✐s ❛ ▼❛r❦♦✈ ❝❤❛✐♥ X 0 , X 1 , ... st❛rt❡❞ ✐♥ ❡q✉✐❧✐❜r✐✉♠✳ ❚♦ tr❡❛t t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥str✉❝t✳ ❉❡✜♥✐t✐♦♥✳ ●✐✈❡♥ ❛ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω , F , P ) ✱ ❛ ♠❡❛s✉r❛❜❧❡ ♠❛♣ T : Ω → Ω ✐s s❛✐❞ t♦ ❜❡ ♣r♦❜❛❜✐❧✐t② � T − 1 A � = P ( A ) ❢♦r ❛❧❧ A ∈ F ✱ ✇❤❡r❡ T − 1 A = { ω ∈ Ω : Tω ∈ A } ❞❡♥♦t❡s t❤❡ ♣r❡✐♠❛❣❡ ♦❢ A ♣r❡s❡r✈✐♥❣ ✐❢ P ✉♥❞❡r T ✳ ❲❡ s❛② t❤❛t t❤❡ t✉♣❧❡ (Ω , F , P, T ) ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♣r❡s❡r✈✐♥❣ ❞②♥❛♠✐❝❛❧ s②st❡♠ ✳ ■t❡r❛t❡s ♦❢ T ❛♥❞ T − 1 ❛r❡ ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② ❜② T 0 ω = ω ❛♥❞✱ ❢♦r n ≥ 1 ✱ T n = T ◦ T n − 1 , T − n = T − 1 ◦ T − ( n − 1) = ( T n ) − 1 . ✯ ❲❡ ✉s❡ t❤❡ ✐♥✈❡rs❡ ✐♠❛❣❡ ✐♥ ♦✉r ❞❡✜♥✐t✐♦♥s ❜❡❝❛✉s❡ A ∈ F ❞♦❡s ♥♦t ♥❡❝❡ss❛r✐❧② ✐♠♣❧② t❤❛t TA ∈ F ✳ ❆❧s♦✱ ❜❡✇❛r❡ t❤❛t s♦♠❡ ❛✉t❤♦rs s❛② t❤❛t ✏ P ✐s ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r T ✑ r❛t❤❡r t❤❛♥ ✏ T ♣r❡s❡r✈❡s P ✳✑ ❋✐♥❛❧❧②✱ ♦❜s❡r✈❡ t❤❛t s✐♥❝❡ t❤❡ ♣✉s❤✲❢♦r✇❛r❞ ♠❡❛s✉r❡ T ∗ P = P ◦ T − 1 ✐s ❡q✉❛❧ t♦ P ✱ t❤❡ ❝❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡s ❢♦r♠✉❧❛ s❤♦✇s t❤❛t ˆ ˆ ˆ f ◦ TdP = fdT ∗ P = fdP Ω Ω Ω ❢♦r ❛❧❧ f ❢♦r ✇❤✐❝❤ t❤❡ ❧❛tt❡r ✐♥t❡❣r❛❧ ✐s ❞❡✜♥❡❞✳ ■❢ X ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ (Ω , F , P ) ❛♥❞ T : Ω → Ω ✐s ♣r♦❜❛❜✐❧✐t② ♣r❡s❡r✈✐♥❣✱ t❤❡♥ X n ( ω ) = X ( T n ω ) ❞❡✜♥❡s ❛ st❛t✐♦♥❛r② s❡q✉❡♥❝❡ s✐♥❝❡ ❢♦r ❛♥② n, k ∈ N ❛♥❞ ❛♥② ❇♦r❡❧ s❡t B ∈ B n +1 ✱ ✐❢ A = { ω : ( X 0 ( ω ) , ..., X n ( ω )) ∈ B } ✱ t❤❡♥ T − k A � � P (( X k , ..., X n + k ) ∈ B ) = P = P ( A ) = P (( X 0 , ..., X n ) ∈ B ) . ■♥ ❢❛❝t✱ ❡✈❡r② st❛t✐♦♥❛r② s❡q✉❡♥❝❡ t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ ❛ ♥✐❝❡ s♣❛❝❡ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤✐s ❢♦r♠✿ ■❢ Y 0 , Y 1 , ... ✐s ❛ st❛t✐♦♥❛r② s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ ❛ ♥✐❝❡ s♣❛❝❡ ( S, S ) ✱ t❤❡♥ t❤❡ ❑♦❧✲ � S N 0 , S N 0 � ♠♦❣♦r♦✈ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠ ❣✐✈❡s ❛ ♠❡❛s✉r❡ P ♦♥ s✉❝❤ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡ ♣r♦❥❡❝t✐♦♥s X n ( ω ) = ω n s❛t✐s❢② ( X 0 , X 1 , ... ) = d ( Y 0 , Y 1 , ... ) ✳ ■❢ ✇❡ ❧❡t X = X 0 ❛♥❞ T = θ ✭t❤❡ s❤✐❢t ♠❛♣✮✱ t❤❡♥ T ✐s ♣r♦❜❛❜✐❧✐t② ♣r❡✲ s❡r✈✐♥❣ ❛♥❞ X n ( ω ) = ω n = ( θ n ω ) 0 = X ( T n ω ) ✳ ✼✻