erv ce station has both self sen ce and full-service IS ands n each sand there i

Question

erv ce station has both self sen ce and full-service IS ands n each sand there i5 Single regular un ace pump with two hoses et X denote the number D hoses being used on the self-service island a denote the number of hoses on the full-service island in use at that time. lhe joint pmf of and y appears in the accompanying tabulation. partial r me, and I pix, y) 0 0.10 0.04 0.01 0.07 0.20 0.07 0.05 0.14 0.31 (a) What is Px - 1 and Y- 1)? = 1 and =1) (b) Compute PX1 and Ys1) (c) Given 2 hoses are being used on the self-service island, what is the probability at least 1 hose is being used at the full-service island? (Use 5 significant digits)- Compute the probability of this event, P(X # 0 and # 0) = (d) Compute the marginal pmf of x. Px(x) Compute the Expected number of hoses baing used on the self service islanod Compute the marginal pmf of Y pty) Compute the Expected number of hoses baing used on the full-service island Given E[xY is 1.8600, compute the Covariancex, r (hint: use shortcut formula; Use 5 significant digits). If VanX is 0.5304 and VarYT is 0.5944, then compute the Correlatonpx,Y (Use 5 significant digits)

Explanation / Answer

joint density of (x,y)

marginal distirbution of x:

marginal distribution of y:

a)

P(X=1,Y=1)=0.20

b)P(X<=1 ; Y<=1) =0.41

c-1)P(Y>=1|X=2) =(0.14+0.31)/0.51=0.88235

c-2) P(Xnot 0 and Ynot 0) =0.9

d)

expected number =1.36

expected numebr =1.16

Cov(X,Y)=E(XY)-E(X)*E(Y)=0.28240

Correlation =Cov(X,Y)/(Var(X)*Var(Y))1/2 =0.50295

y x 0 1 2 Total 0 0.1000 0.0400 0.0100 0.1500 1 0.0700 0.2000 0.0700 0.3400 2 0.0600 0.1400 0.3100 0.5100 Total 0.2300 0.3800 0.3900 1.0000

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