From a considerable experience the campus parking authority has devel- oped the

Question

From a considerable experience the campus parking authority has devel- oped the following probability distribution of the number of tickets issued per semester to students with parking

permits

number of tickets 0 1 2 probability 0.6 0.3 0.1

a) Suppose that each ticket costs $50. Find the expected cost and the standard deviation of the cost to the permit holder per semester.

1

b) Suppose that two permit holders are selected at random (this means that the number of tickets issued to one permit holder is independent of the number of tickets issued to the other permit holder).

Let X = the number of tickets received by the i’th ticket holder, i=1,2. c) Find the joint probability distribution Xof 1 and 2
d) Let t =X1+ X2What is the expected total number of tickets and

the standard deviation of the total number of tickets they receive?
e) What is the expected total cost and the standard deviation of the

total cost to the two permit holders?

Explanation / Answer

Ans:

a)Expected cost=50*(0.6*0+0.3*1+0.1*2)=50*(0+0.3+0.2)=50*0.5=25

variance=(0-0.5)2*0.6+(1-0.5)2*0.3+(2-0.5)2*0.1=0.45

standard deviation=50*sqrt(0.45)=50*0.671=33.54

b)Joint probability distribution

d)pmf for X1+X2:

Expected number of tickets=(0*0.36+1*0.36+2*0.21+3*0.06+4*0.01=(0.36+0.42+0.18+0.04)=1

standard deviation of number of tickets=sqrt[(0-1)2*0.36+(1-1)2*0.36+(2-1)2*0.21+(3-1)2*0.06+(4-1)2*0.01]

=sqrt(0.36+0+0.21+0.24+0.09)

=sqrt(0.9)

=0.95

e)Expected cost=50*1=50

standard deviation=50*0.95=47.5

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