Math 375 Archives - Page 14 of 15 - Shawn Zhong

$What does a proof look like? • Assumptions • Conclusion • Proof Example 1 • Assumption ○ V={(x_1,x_2,x_3 )|x_1,x_2,x_3∈Rand x_1+x_3=0} ○ ∀ x,y∈V, x+y is defined by ○ z=x+y if z=(x_1+y_1,x_2+y_2,x_3+y_3 ) ○ tx is defined by tx=(tx_1,tx_2,tx_3 ) for every x∈V,t∈R • Conclusion ○ V is a vector space • Proof: Axiom 1 (∀x,y∈V:x+y∈V) ○ let z=(z_1,z_2,z_3 )=x+y=(x_1+y_1,x_2+y_2,x_3+y_3 ) ○ z_1+z_3=x_1+y_1+x_3+y_3=(x_1+x_3 )+(z_1+z_3 )=0 ○ ⇒z∈V Example 2 • Assumption ○ V={(x_1,x_2,x_3 )|x_1,x_2,x_3∈Rand x_1+x_3=1} ○ ∀ x,y∈V, x+y is defined by ○ z=x+y if z=(x_1+y_1,x_2+y_2,x_3+y_3 ) ○ tx is defined by tx=(tx_1,tx_2,tx_3 ) for every x∈V,t∈R • Conclusion ○ V is not a vector Space • Proof: ∃x,y∈V:x+y∉V Axiom 5 • To show Axiom 5 does not hold, • we have to prove for every O∈V, • there is an x∈V with O+x≠x Example 3 • Assumption ○ V={all functions f:[0,1]→R} • Conclusion ○ V is a vector space • Proof: Axiom 3（∀f,g∈V:f+g=g+f） ○ Let h=f+g and k=g+f ○ Both h and g has a domain of [0,1] ○ h(x)=f(x)+g(x)=g(x)+f(x)=k(x)$

Math 431 – 9/11

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