The document discusses indicator random variables and how they can be used to analyze a hiring problem algorithm. An indicator random variable is defined as a random variable associated with a specific event that takes the value 1 if the event occurs and 0 if it does not. The expected value of an indicator random variable is equal to the probability of the associated event. The hiring problem involves interviewing candidates each day and hiring the best one seen so far. Indicator random variables can be used to represent whether each candidate is hired, allowing the expected number of hires to be calculated.

1. Indicator Random Variable
Raditya W Erlangga (G651120714)
Bogor, 15 Desember 2012

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

3. Hiring Problem
HIRING NEW OFFICE ASSISTANT
Optimal strategy to maximize the
probability of selecting
best applicant
SECURITY UPDATE
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4. 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?

5. HIRING CANDIDATE ALGORITHM
HIRE-ASSISTANT (n)
1. best = 0 // candidate 0 is a least-qualified dummy candidate
2. for i = 1 to n
3. interview candidate i
4. If candidate i is better than candidate best
5. best = i
6. hire candidate i
Suppose:
ci = interview process
ch = hiring process
The complexity is O(ci n + chm)

6. 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:

7. 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

8. 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

9. 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:

10. 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

11. 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]:

12. QUESTIONS?

13. THANK YOU