M20worksheet 4 answers PDF

Preview

Summary

Lecture notes for lecture 10...

Description

+∞ ∑

kP (X = k).

k=1

+∞ ∑

P (X ≥ k).

k=1

|

{z

∑

}

P (X ≥ k)

k=1

∑ (P (X = k) + P (X = k + 1) + P (X = k + 2) + · · · ). k=1

∑

P (X ≥ k) =

k=1

+∞ ∑

= =

k=1 +∞ ∑

k=1 +∞ ∑

∞ ∑

kP (X = k) = E[X]

k=1

P (X ≥ k) (1 − p)k−1 (1 − p)l

l=0

=

1 1 = 1 − (1 − p)

 mX (x) 

x · P (X = x|A). x∈Ω

k ∑ j=1

E[X|Fj ]P (Fj )....