Question

Four buses carrying 100 students from the same school arrive at a football stadium. The busses carry, respectively, 10, 20, 30, and 40 students. One of the student is randomly selected. Let X denote the number of students who were on the bus carrying the randomly selected student. Let Y denote the number of students on the bus when one of the 4 bus drivers is randomly selected.

(a) Explain (without any computation) which of Var(X) or Var(Y ) you think is larger? Why?
(b) Compute E[X] and E[Y].

Answer 1

a) Var (x) is larger

b)E(x)= 10 and E(y)= 2.5

Explanation:

a) Var (x) is larger because the number of students selected can be greater than 1 but Var(y) will always be less because the numbers of drivers chosen can be only one at a time.

b) The expected value of X and y can be computed by multiplying the numbers with their probabilities

x 10 20 30 40

P(x) 0.1 0.2 0.3 0.4

P (x= 10) = 10/ 10+20+30+40= 10/ 100= 0.1

E(x) = 10(0.1) +20(0.2) +30(0.3)+40(0.4)

E(x)= 1=2+3+4= 10

Now the expected value of y

y 1 2 3 4

P(y) 0.25 0.25 0.25 0.25

E(y)= 1(0.25) +2(0.25) +3(0.25)+4(0.25)

E(y)= 0.25 +0.5 +0.75 +1 = 2.5