Delhi School of Economics Course 003: Basic Econometrics
July 2016
Rohini Somanathan
Problem Set 1
(due during tutorials the week starting August 22nd)
1. Prove the following results using only logical arguments (with no algebra or computation) (a)
n k
!
= n
n−k
!
(b)
n n−1 k−1
!
=k n k
!
(c)
m+n k
!
=
k
X
j=0
m j
! n k−j
!
2. Can you construct a simple example of 3 events, A, B and C which are dependent even though P(A∩B∩C) =P(A)P(B)P(C)? (I’m obviously not looking for the one in your slides)
3. A family has two children and it is known that at least one is a girl. What is the probability that both are girls, given this information? What if it is known that the elder child is a girl?
4. Show that the two Hypergeometric random variables, X ∼HGeom(w, b, n) andY ∼HGeom(n, w+ b−n, w) have the same distribution. Once again, you should be able to do this using the story of double-tagged items that underlies the Hypergeometric distribution.
5. There are 100 slips of paper in a hat, each with a different number between 1 and 100.
(a) Five slips are drawn with replacement. What is the distribution of the number of drawn slips with the value of at least 80 written on them? What is the probability that the number 100 is drawn at least once?
(b) Now suppose these five slips are drawn without replacement. What is the distribution of the number of drawn slips with the value of at least 80 written on them? What is the distribution of the value of the jth draw for 1 ≤ j ≤ 5? What is the probability that the number 100 is drawn at least once? Is this higher or lower than for the case of drawing with replacement?
Why?
