2017 - Page 21 of 43 - Shawn Zhong

Math 431 – 9/11

$Probability Space • Ω: sample space (list of all outcomes) • F: collection of events (subsets of Ω) • P: probability measure ○ P(A)∈[0,1] ○ P(∅)=0 ○ P(Ω)=1 ○ For disjoint A_1,A_2,…:P(⋃24_(i=1)^∞▒A_i )=∑_(i=1)^∞▒PA_i ) Equally Likely Outcome • P(ω)=1/(#Ω),∀ω∈Ω • P(A)=(#A)/(#Ω) • Example: 431 game with full deck ○ Ω={(c_1,c_2,c_3 )│■8(c_1 is my card@c_2 is your first card@c_3 is your second card@and they are all distinct)} ○ P(A)=(#A)/(#Ω) ○ W_7={(c_1,c_2,c_3 )∈Ω| (c_2≥7 and c_2c_1 ) or (c_27 and c_3c_1 )} ○ #Ω=52×51×50=(52)_3 ○ Note: (n)_k=n!/(n−k)! • Example: 431 game with replacement ○ Ω={(c_1,c_2,c_3 )│■8(c_1 is my card@c_2 is your first card@c_3 is your second card)} ○ W_7={(c_1,c_2,c_3 )∈Ω| (c_2≥7 and c_2c_1 ) or (c_27 and c_3c_1 )} ○ #Ω=〖52〗^3 Different Types of Random Experiments • S={1,…,n} • Sampling with replacement where order matters ○ Ω=S^k={(s_1,…,s_k )|s_i∈S} ○ #Ω=n^k • Sampling without replacement where order matters ○ Ω={(s_1,…,s_k )|s_i∈S and ∀i≠j:s_i≠s_j } ○ #Ω=n(n−1)⋯(n−k+1)=n!/(n−k)!=(n)_k • Sampling without replacement where order is irrelevant ○ Ω={A⊆S|#A=k}$

Math 375 – 9/11

Field • A field F is a set together with 2 binary operations • +, × (− optional) that satisfies the following: ○ a+b=b+a ○ (a+b)+c=a+(b+c) ○ a×b=b×a ○ (a×b)×c=a×(b×c) ○ a×(b+c)=a×b+a×c ○ There is a special element O, such that a+O=a ○ There is a special element 1, such that 1×a=a ○ For all a, there is a b, such that a+b=0 ○ For any a≠O, there is a b, such that a×b=1 ○ Optional:1≠O, O≠1 • Example ○ F={0,1} ○ +≔{█(0+0=0@0+1=1@1+1=0)┤ ○ ×≔{█(0×0=0@0×1=0@1×1=1)┤ • Example ○ F={0,1,2} ○ +≔{█(0+0=0@0+1=1@0+2=2@1+1=2@1+2=0@2+2=1)┤ ○ ×≔{█(0×0=0@0×1=0@0×2=0@1×1=1@1×2=2@2×2=1)┤ Vector Space • A vector space V(over F) is a set together with binary operations • {█(+:V+V→V@×:F×V→V)┤, such that ○ F is a field ○ u+v=v+u, ∀u,v∈V ○ (u+v)+w=v+(u+w), ∀u,v,w∈V ○ There is a 0 and vector 0 ⃗, such that § ∀u,v∈V, ∀a,b∈F § u+0 ⃗=u § 0×u=0 ⃗ § a×0 ⃗=0 ⃗ § (a×b)×u=a×(b×u) § (a+b)×u=a×u+b×u § a(u+v)=a×u+a×v § u+(−1)u=(1+(−1))×u=0×u=0 ⃗

Math 375 – Homework 1

Math 375 – 9/7

$Linear Space / Vector Space • A set of vectors • A set of numbers • Addition of vectors • Multiply vectors with numbers Zero Vector • There is a vector O such that for all vector x ○ x+O=x • Theorem ○ If O_1 and O_2 are both zero vectors, then O_1=O_2 • Proof ○ {█(O_1+O_2=O_1@O_2+O_1=O_2 )┤⇒O_1=O_2 Existence of Negative Vector • For every vector x, there is a vector y such that • x+y=0 • denoted as −x Multiplication with Numbers (Scalers) • x,y:vectors, s,t:numbers (Number field:Q,R,ℂ) • s(x+y)=sx+sy • (s+t)x=sx+tx • s(tx)=(st)x • 0⋅x=0 • 1⋅x=x Example of a Common Vector Spaces • R3={(x_1,x_2,x_3 )│x_1∈Rx_2∈Rx_3∈R is a vector space • Addition and multiplication defined as ○ (x_1,x_2,x_3 )+(y_1,y_2,y_3 )≝(x_1+y_1,x_2+y_2,x_3+y_3 ) ○ t(x_1,x_2,x_3 )≝(tx_1,tx_2,tx_3 ) Example of a Strange Vector Spaces • Number:R • Vector:R+=(0,∞) • Addition ○ x⨁y=x×y ○ e.g. √2⨁√2=√2×√2=2 ○ Zero vector: 1 • Inverse of Addition ○ Given x, find y ○ x⨁y=1 ○ ⇒y=1/x • Multiplication with numbers ○ t∈R, x∈R_+ ○ t⨀x≝x^t • Proof: Distributive law ○ t⨀(s⨀x)=(x^s )^t=x^st=(ts)⨀x$