Question

our buses carrying 140 high school students arrive to Montreal. The buses carry, respectively, 31, 43, 27, and 39 students. One of the studetns is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on his bus. Compute the expectations and variances of X and Y:

Answer 1

Answer:

V(X)= 39.10

V(Y)= 40

Step-by-step explanation:

Given that

Total number of student = 140

Bus A - 31

Bus B- 43

Bus C- 27

Bus D- 39

The probability that a student was on the bus is proportional to the number of student. Eg 31/140 in bus A, 43/140 on bus B, ...

E(X) = (31*31/140) + (43*43/140)  + (27*27/140) + (39*39/140)

=  35.5

$Var(X) =(31-35.5)^2 \times \dfrac{31}{140}+(43-35.5)^2 \times \dfrac{43}{140}+(39-35.5)^2 \times \dfrac{39}{140}+(27-35.5)^2 \times \dfrac{27}{140}$

V(X)= 39.10

The bus driver have  1/4 probability on being on any of the buses.

E(Y) = 140/4 = 35

$Var(Y)=\dfrac{(35-31)^2+(35-43)^2+(35-27)^2+(35-39)^2}{4}$

V(Y)= 40