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

Question

n 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 hospitals emergency room has tour beds, whereas the north hospital's emergency room has only three beds. Let X denote the nur ber or south beds occupe at d pdf cular tirme on d Q ven day, dn lec denute the ru be of north beds occupied dt he sa ne ne on the san e day. Suppose that these two r s are "dependent; that he p nof puts p ability masses 0 2 0 2 0.3 0.1 and 0.2 on the x values 0. 1, 2. 3, and 4, respectively, and that the Pmf of distributes probabilities o.1,0·2,0.3, and 0.4 on the y values U, 1, 2, and 2, respectively. [a Display the joint prmr or x and in a joint probability table. 0.2 0.3 0.4 0.3 0.1 0.2 (b) 1 and 1) by adding probabilities from the joint pmf, and verify that this equals the product cf,#x s 1} and yg 1). Compute ror Pxs1 and Ys 1) ) Express the event that the total number of bads occupiad at the two hospitals combined is at most 1 in terms cfxand Y {x, Y: K.YCI) Cakulata this prohabiity. d) What is the prebability that at least one of uhe two huspitals hes no bes oxup ed?

Explanation / Answer

for abvoe to be independent :P(x,y) =P(x)*P(Y)

hence probability in any cell =P(column sum)*P(row sum)

b)

P(X<=1 and Y<=1) =0.12

c)

{X,Y:X+Y<=1}

probability=P(X=0;Y=0)+P(X=1;Y=0)+P(X=0;Y=1)=0.02+0.02+0.04 =0.08

d)

Probability =0.2+0.1-0.02 =0.28

p(x,y) 0 1 2 3 4 total 0 0.02 0.02 0.03 0.01 0.02 0.1 1 0.04 0.04 0.06 0.02 0.04 0.2 2 0.06 0.06 0.09 0.03 0.06 0.3 3 0.08 0.08 0.12 0.04 0.08 0.4 total 0.2 0.2 0.3 0.1 0.2 1

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