indicator random variables

Raditya W Erlangga (G651120714)Bogor, 15 Desember 2012

Indicator Random Variable

AGENDA

• Introduction• Hiring Problem overview• Indicator Random Variable• Examples• Q & A

SECURITY UPDATE

<ISC SA or IR Number>

<Date>

HIRING NEW OFFICE ASSISTANTOptimal strategy to maximize the

probability of selecting best applicant

Hiring Problem

HIRING PROBLEM OVERVIEW

» A new office assistant is required» Using employment agency» Interviews the candidate each day» A small fee required to pay the interview process done by the agency.

However, hiring an applicant is more expensive: pay substantial hiring fee + fire current office assistant to get the best candidate for the job

» What is the price needed for this strategy?

HIRING CANDIDATE ALGORITHM

HIRE-ASSISTANT (n)1. best = 0 // candidate 0 is a least-qualified dummy candidate2. for i = 1 to n3. interview candidate i4. If candidate i is better than candidate best5. best = i6. hire candidate i

Suppose:ci = interview process

ch = hiring process

The complexity is O(ci n + chm)

INDICATOR RANDOM VARIABLES

» Is used to analyze the hiring problem algorithm» a convenient method for converting between probabilities and

expectations» Given a sample space S and an event A, the indicator random variable

I{A} associated with event A is defined as:

EXAMPLES

» Flipping a fair coin» Sample space S = { H,T }, Pr{H} = Pr{T} = 1/2» Define indicator random variable XH associated with the coming up

heads:

with H as the event

LEMMA 1

Given a sample space S and an event A in the sample space S, let XA =I{A}.

Then E [XA ] = Pr{A}

Proof:E [XA ] = E[I{A}]

= 1. Pr{A} + 0. Pr{A’} = Pr{A}

where A’ is S – A, the complement of A

EXPECTED VALUES OF EVENT

» Xi = I {the ith flip results in the event H}» X = random variable denoting the total number of heads in the n coin

flips

» The expected number of event H:

ANALYSIS OF THE HIRING PROBLEM USING INDICATOR RANDOM VARIABLES

» Assume the candidates arrive in random order» X = random variable whose value equals the number of times we hire

a new office assistant» We may use expected value of random variable equation:

» However, a simplified calculation is using indicator random variables.» Instead of computing E[X] by defining one variable associated with the

number of times we hire a new office assistant, define n variables related to whether or not each particular candidate is hired. In particular, let Xi be the indicator random variable associated with the event in which the ith candidate is hired

and

ANALYSIS OF THE HIRING PROBLEM USING INDICATOR RANDOM VARIABLES» Based on lemma E [XA ] = Pr{A}, we have:

» Since the candidate i arrives in random order, any one of these first i candidates is likely to be best-qualified candidate so far. Thus, the probability of candidate i is 1/i better qualified than candidates 1 till i-1, which yields:

» Now we can compute E[X]:

QUESTIONS?

THANK YOU