Question

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation. 

$$ \begin{array}{lc|ccc}  & & & y & \\ p(x, y) & & 0 & 1 & 2 \\ \hline & 0 & 0.10 & 0.03 & 0.01 \\ x & 1 & 0.06 & 0.20 & 0.06 \\ & 2 & 0.06 & 0.14 & 0.34 \end{array} $$

 (a) Given that X = 1, determine the conditional pmf of Y-i.e., pyix(011), pYix(111), pyx(211). (Round your answers to four decimal places.) 

(b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

 (c) Use the result of part (b) to calculate the conditional probability P(Ys 1| X= 2). (Round your answer to four decimal places.) 

(d) Given that two hoses are in use. the full-service island, what is the conditional pmf of the number in use at the self-service island? (Round your answers to four decimal places.)

Answer 1

a)

y	0	1	2
P(y|1)	0.1875	0.6250	0.1875

b)

y	0	1	2
P(y|2)	0.1111	0.2593	0.6296

c) P(Y<=2|X=2)=0.1111+0.2593 =0.3704

d)

x	0	1	2
P(x|2)	0.0244	0.1463	0.8293