Question

Find the cumulative distribution function of X and draw its graph?

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability 0.3, and his second appointment will lead independently to a sale with probability 0.6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500. X is the total dollar value of all sales. Hint: you could find the probability mass function of X and use that.

Answer 1

here below is the probability mass function of X:

P(X=0)=P(no sales from both of appointments)=(1-0.3)*(1-0.6)=0.28

P(X=500)=P(no sales frm first appointment and standard model sale from second appointment)+P(no sales frm second appointment and standard model sale from first appointment)

=(1-0.3)*0.6*(1/2)+0.3*(1/2)*(1-0.6)=0.27

P(X=1000)=P(no sales frm first appointment and deluxe model sale from second appointment)+P(no sales frm second appointment and deluxe model sale from first appointment)+P(standard sale from first appointment and standard sale from second appointment)

=(1-0.3)*0.6*(1/2)+0.3*(1/2)*(1-0.6)+0.3*(1/2)*0.6*(1/2)=0.315

P(X=1500)=P(standard sale from first appointment and deluxe from second appointment)+P(deluxe sale from first appointment and standard sale from second appointment)

=0.3*(1/2)*0.6*(1/2)+0.3*(1/2)*0.6*(1/2)=0.09

P(X=2000)=P(deluxe sale from first appointment and deluxe from second appointment)

=0.3*(1/2)*0.6*(1/2)=0.045

hence from above below is CDF of X:

	0		x<0
	0.28		0<=x<500
	0.55		500<=x<1000
F(x)=	0.865	for x=	1000<=x<1500
	0.955		1500<=x<2000
	1		x>=2000

graph of F(x):