In: Statistics and Probability
Consider flipping a coin 5 times, and define the following
events:
A = "The very first flip is tails"
B ="The first three flips are tails"
C = "The middle (third) flip is tails"
D = "The last three flips are tails"
E = "The very last flip is tails"
1)Which of the following collections of events (if any) are pairwise independent?
a) {A,B,C}
b) {B,C,D}
c) {B,D,E}
d) None
2) Which of the following pairs of events (if any) are
conditionally independent, given the
third set?
a) {A,B} given C
b) {B,C} given E
c) {B,D} given C
d) None
ANSWERS..
1) d) none
2) c) {B,D} given C
Solution is given as hand written images below
Lets look at each cases
1)
a) {A,B,C}
P(A) = 1/2 P(B)= 1/8 P(C) = 1/2
P(AB)=1/8
Clearly P(AB) not equal to P(A)*P(B)
So, {A,B,C} is not pairwise independent
b) {B,C,D}
P(BC) = 1/8
Clearly P(BC) not equal to P(B)*P(C)
So, {B,C,D} is not pairwise independent
c) {B,D,E}
P(BD)= 1/32 P(D)= 1/8
Clearly P(BD) not equal to P(B)*P(D)
So, {B,D,E} is not pairwise independent
ANSWER IS OPTION d) NONE
2)
a) {A,B} given C
P(AB/C) = 1/4 P(A/C)= 1/2 P(B/C) = 1/4
P(AB/C) is not equal to P(A/C)* P(B/C)
{A,B} given C is not conditionally independent
b) {B,C} given E
P(BC/E) = 1/8 P(B/E)= 1/8 P(C/E)= 1/2
{B,C} given E is not conditionally independent
c) {B,D} given C
P(BD/C)= 1/16 P(B/C)= 1/4 P(D/C)= 1/4
Clearly we can see P(BD/C)= P(B/C)*P(D/C)
So answer is option c) {B,D} given C
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