Question

In: Statistics and Probability

Consider flipping a coin 5 times, and define the following events: A = "The very first...

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

Solutions

Expert Solution

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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