Probability Refresher - Discrete Math - Lecture Slides, Slides of Discrete Mathematics

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Introduction to Discrete

Mathematics

Probability Refresher

What’s a Random Variable?

A Random Variable is a real-valued function on a sample space S

E[X+Y] = E[X] + E[Y]

Random Walks

Lecture 12 (October 10, 2007)

How to walk

home drunk

Abstraction of Student Life

Like finite automata, but instead of a determinisic or non-deterministic action, we have a probabilistic action

Example questions: “What is the probability of reaching goal on string Work,Eat,Work?”

No new ideas

Solve HW problem

Eat

Wait

Work

Work

Hungry

Simpler:

Random Walks on Graphs

At any node, go to one of the neighbors of the

node with equal probability

Simpler:

Random Walks on Graphs

At any node, go to one of the neighbors of the

node with equal probability

Simpler:

Random Walks on Graphs

At any node, go to one of the neighbors of the

node with equal probability

0 n

k

Random Walk on a Line

You go into a casino with $k, and at each time step, you bet $1 on a fair game

You leave when you are broke or have $n

Question 1: what is your expected amount of

money at time t?

Let Xt be a R.V. for the amount of $$$ at time t

0 n

k

Random Walk on a Line

You go into a casino with $k, and at each time step, you bet $1 on a fair game

You leave when you are broke or have $n

Xt = k + δ 1 + δ 2 + ... + δt,

(δi is RV for change in your money at time i)

So, E[X (^) t] = k

E[δi] = 0

Random Walk on a Line

Question 2: what is the probability that you

leave with $n?

E[Xt] = k

E[Xt] = E[Xt| Xt = 0] × Pr(Xt = 0)

• E[Xt | Xt = n] × Pr(Xt = n)
• E[ X (^) t | neither] × Pr(neither)

As t →∞, Pr(neither) → 0, also something (^) t < n

Hence Pr(Xt = n) → k/n

k = n × Pr(Xt = n)

• (somethingt) × Pr(neither)

0 n

k

Another Way To Look At It

You go into a casino with $k, and at each time step, you bet $1 on a fair game

You leave when you are broke or have $n

Question 2: what is the probability that you

leave with $n?

= probability that I hit green before I hit red

Random Walks and

Electrical Networks

Same as equations for voltage if edges all have same resistance!

p (^) x = Pr(reach green first starting from x)

pgreen = 1, p (^) red = 0

And for the rest p (^) x = Average (^) y ∈ Nbr(x) (p (^) y )

0 n

k

Another Way To Look At It

You go into a casino with $k, and at each time step, you bet $1 on a fair game

You leave when you are broke or have $n

Question 2: what is the probability that you

leave with $n?

voltage(k) = k/n = Pr[ hitting n before 0 starting at k] !!!