Question

Consider a small ferry that can accommodate cars and buses. The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. The joint pmf of X and Y is given in the table below.

p(x, y)		y
		0	1	2
x	0	0.150	0.060	0.090
	1	0.050	0.020	0.030
	2	0.100	0.040	0.060
	3	0.125	0.050	0.075
	4	0.025	0.010	0.015
	5	0.050	0.020	0.030

It is readily verified that X and Y are independent.

(a)

Compute the expected value, variance, and standard deviation of the total number of vehicles on a single trip. (Round your variance and standard deviation to two decimal places.)

expected value  vehicles 2.75 correct

variance

standard deviation vehicles

(b)

If each car is charged $3 and each bus $10, compute the expected value, variance, and standard deviation of the revenue resulting from a single trip. (Round your variance and standard deviation to two decimal places.)

expected value$

variance   

standard deviation$

Answer 1

marginal distribution of x
x	P(x)	xP(x)	x-E(x)	(x-E(x))^2	(x-E(x))^2*p(x)
0	0.300	0.000	-1.950	3.803	1.141
1	0.100	0.100	-0.950	0.903	0.090
2	0.200	0.400	0.050	0.003	0.001
3	0.250	0.750	1.050	1.103	0.276
4	0.050	0.200	2.050	4.203	0.210
5	0.100	0.500	3.050	9.303	0.930
total	1.000	1.950			2.648
E(x)	=				1.9500
Var(x)=	E(x^2)-(E(x))^2=				2.6475

marginal distribution of y
y	P(y)	yP(y)	y-E(y)	(y-E(y))^2	(y-E(y))^2*p(y)
0	0.500	0.000	-0.800	0.6400	0.320
1	0.200	0.200	0.200	0.0400	0.008
2	0.300	0.600	1.200	1.4400	0.432
total	1.000	0.800			0.760
E(y)	=				0.8000
Var(y)=σy=	E(y^2)-(E(y))^2=				0.7600
standard deviation =σy =					0.8718

a)expected value =E(X)+E(Y) =1.95+0.8 =2.75

variance =Var(x)+Var(y) =2.6475+0.76 =3.41

standard deviation =1.85

b)

expected value$ =3*1.95+10*0.8=13.85

variance =3^2*2.6475+10^2*0.76=99.83

standard deviation =sqrt(99.83)=9.99