1 Exception
1.1 Definition:
- If R is a random variable defined on a sample space S, then the expectation of R is:
Ex[R]::=∑w∈SR(ω)Pr[ω] - Alternate Definition for any random variable R:
Ex[R]=∑x∈range(R)x⋅Pr[R=x]
1.2 Conditional Expectation
- The conditional expectation Ex[R|A] of a random variable R given event A is:
Ex[R|A]::=∑r∈range(R)r⋅Pr[R=r|A]
1.3 Law of Total Expectation
- Let R be a random variable on a sample space S, and suppose that A1, A2, … , is a partition of S. Then
Ex[R]=∑iEx[R|Ai]⋅Pr[Ai]
1.4 Mean Time to Failure
- If a system independently fails at each time step with probability p, then the expected number of steps up to the first failure is 1p.
1.5 geometric distribution
- A random variable, C , has a geometric distribution with parameter p iff codomain(C)=Z+ and:
Pr[C=i]=(1−p)i−1p - If a random variable C has a geometric distribution with parameter p, then:
Ex[C]=1p
2 Linearity of Expectation
- For any random variables R1 and R2,
Ex[a1R1+a2R2]=a1Ex[R1]+a2Ex[R2] - Sums of Indicator Random Variables:
Linearity of expectation is especially useful when you have a sum of indicator random variables
2.1 Expectation of a Binomial Distribution
- pn
2.2 Expectations of Products
- If random variables R1,R2,...,Rkare mutually independent, then
E(∏i=1kRi)=∏i=1kEx[Ri]
Reference
[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.