CS513- Midterm Exam- Solved

#1 (10 Points)

Is the following function a proper distance function?  Why?  Explain your answer.



Hint: Measure the distance between (0,0), (0,1) and (1,1)

Ans:

-        For any function to be a proper distance function, it should satisfy below conditions:

1.     Distance should be non-negative

2.     For two points, distance should be commutative

3.     Distance should hold triangle inequality,

i.e.  for a,b,c, d(a,c) <= d(a,b) + d(b,c)

 

-        Let’s check these conditions on a=(0,0), b=(0,1) and c=(1,1)

1.     First condition satisfy directly, as given distance function is a squared difference

2.     d(a,b) = (0-0)2 + (0-1)2 = 0 + 1 = 1

d(b,a) = (0-0)2 + (1-0)2 = 0 + 1 = 1

Hence, 2nd condition satisfies

3.     d(a,c) = ((0-1)2 + (0-1)2)2 = 4

d(a,b) = 1

d(b,c) = ((0-1)2 + (1-1)2)2 = 1

d(a,c) = 4 2 = 1 + 1 = d(a,b) + d(b,c)

Hence, d(a,c) d(a,b) + d(b,c) and 3rd condition fails.

-        So, given function is not a proper distance function.


 

# 2 (15 Points)

A large department store sells sport shirts in three sizes (Small, Medium and Large), three patterns (plaid/Pl, print/Pr, and stripe/Sr), and two sleeve lengths (long and short).  The accompanying tables give the proportions of shirts sold falling in the various category combinations.

·      What is the probability that the next shirt sold is a medium long-sleeved, print shirt?  Why?

Ans:                        P(medium ∩ long-sleeved ∩ print-shirt ) = 0.05

 

From the given table, to find given probability, first we will go to long-sleeved table then we can just look at an intersection of medium and print-shirt to get the given probability.

 

·      What is the probability that the next shirt sold is a medium print shirt?  Why?

Ans:                        P(medium ∩ print-shirt | short-sleeved) +

P(medium ∩ print-shirt | long-sleeved) =  0.07 + 0.05 = 0.12

 

·      What is the probability that the next shirt sold is a short sleeved shirt?   A long-sleeved shirt?  Why?

Ans:            P(short-sleeved)

= P(small | short-sleeved) + P(medium| short-sleeved) + P(large | short-sleeved)

                  = P(Pl | small) + P(Pr | small) + P(Sr | small) +

                     P(Pl | medium) + P(Pr | medium) + P(Sr | medium) +

                     P(Pl | large) + P(Pr | large) + P(Sr | large)

                  = (0.04 + 0.02 + 0.05) + (0.08 + 0.07 + 0.12) + (0.03 + 0.07 + 0.08)

                  = 0.11 + 0.27 + 0.18

                  = 0.56

 

                  P(long-sleeved) = 1 – P(short-sleeved) = 1 – 0.56 = 0.44

                 

·      Given that the shirt just sold was a short sleeved, plaid, what is the probability that its size was medium?

Ans:            P(short-sleeved ∩ medium | short-sleeved, plaid )

                  = P(short-sleeved ∩ medium) / P(short-sleeved ∩ plaid)

                  = 0.08 / (0.04+0.08+0.03)

                  = 0.08 / 0.15

                  = 0.53

 

 

·      Given that the shirt just sold was medium, plaid, what is the probability that it was short sleeved?  Long-sleeved?

Ans:            P(short-sleeved | medium, plaid )

                  = P(short-sleeved ∩ medium ∩ plaid) / P(medium ∩ plaid)

                  = 0.08 / (0.08+0.1)

                  = 0.08 / 0.18

                  = 0.44

 

P(long-sleeved | medium, plaid )

                  = P(long-sleeved ∩ medium ∩ plaid) / P(medium ∩ plaid)

                  = 0.1 / (0.08+0.1)

                  = 0.1 / 0.18

                  = 0.55

 

 


 

#3 – A) – 1: K = 2 and method = ” unweighted vote” is used

 



#3 – A) – 2: K = 3 and method = ” distance weighted vote” is used

Q-3-B : K=3, distance weighted voting: