A large but sparsely populated county has two small hospitals, one at the south

Question

A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end. The south hospital's emergency room has four beds, whereas the north hospital's emergency room has only three beds. Let X denote the number of south beds occupied at a particular time on a given day, and let Y denote the number of north beds occupied at the same time on the same day. Suppose that these two rv's are independent; that the pmf of X puts probability masses 0.3, 0.1, 0.2, 0.2, and 0.2 on the x values 0, 1, 2, 3, and 4, respectively; and that the pmf of Y distributes probabilities 0.3, 0.1, 0.2, and 0.4 on the y values 0, 1, 2, and 3, respectively.

(a) Display the joint pmf of X and Y in a joint probability table p(x, y) 0 2 3 4 0.3 0.1 0.2 0.4 0 Y 1 2 3 0.3 0.1 0.2 0.2 0.2

Explanation / Answer

a) for x and y are independent ; tehrefore cell probability=row probability *column probability

hence below is joint probability:

b)

P(X<-1 and Y<=1)=P(0,0)+P(0,1)+P(1,0)+P(1,1) =0.160

c){X,Y,X+y<=1}

probability=P(0,0)+P(0,1)+P(1,0) =0.150

d)P(at least one of the hospitals has no bed) =P(X=0)+P(Y=0)-P(X=0,Y=0)=0.3+0.3-0.09 =0.510

y x 0 1 2 3 4 Total 0 0.090 0.030 0.060 0.060 0.060 0.300 1 0.030 0.010 0.020 0.020 0.020 0.100 2 0.060 0.020 0.040 0.040 0.040 0.200 3 0.120 0.040 0.080 0.080 0.080 0.400 Total 0.300 0.100 0.200 0.200 0.200 1.000

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