Question

Additional Exercise 12.7.6. A family with a mother, father, two daughters and three sons lines up in a random order for a photo.

(a) Let D be the random variable denoting the number of daughters who are standing next to the mother and for i = 1,2 let Di be the indicator variable that is 1 if daughter i is next to the mother and 0 otherwise. What is the relationship betweenD, D1, and D2?

(b) What are the expectations E[D1] and E[D2]?

(c) Use linearity of expectation to compute E[D].

(d) Let N be the random variable denoting the number of sons who are standing next to the mother. Use appropriate indicator variables together with linearity of expec- tation to find E[N]. State where you are using linearity of expectation.

Answer 1

solution:

A).

D = D_{1 +} D₂

B)

Total number of arrangements = 7!

Number of cases i th daughter is next to the mother

= 6! × 2!

(consider mother i th daughter as a single entity and arrange the family. then permut mother and i th daughter among themselves)

p( D_i =1) =

6! × 2! / 7! = 2/7

P( =0) = 1 - (2/7) =5/7

E() = =( 0× 5/7 )+(1×2/7) = 2/7

C)

E(D) = E() = E() +E( ) = 4/7

D)

N = Number of sons standing next to the mother

be the indicator variable whwhich is 1 if i th son is next to the mother and 0 otherwise

Clearly N =

+ +

P( =1) = 2/7

p( =0) = 5/7

E(N_i) = 2/7

E(N) = E(N1) +E(N2) +E(N3)=6/7

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