When an automobile is stopped by a roving safety patrol, each tire is checked fo

Question

When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires. (a) ItX and Y are independent with px(0) = 0.5, px(1)-0.3, px(2)-0.2, and pr(0) : 0.2, pr(1)-0.1, py(2) = py(3) = 0.05, pr(4) = 0.6, display the joint pmf of (X, Y) in a joint probability table. (x, y) 0 2 3 4 0 2 b) Compute P(X s 1 and Y S 1) from the joint probability table. P(X 1 and Y 1) = Does P(X S 1 and Y S 1) equal the product P(X S 1) P(Y S 1)? O Yes O No (c) what is P(X + Y = 0) (the probability of no violations)? (d) Compute P(X + 1)

Explanation / Answer

as x and Y are independent therefore P(x,y)=P(x)*P(y)

hence a)

b) P(X<=1 and Y <=1)=P(0,0)+P(0,1)+P(1,0)+P(1,1)=0.24

here P(X<=1)=0.5+0.3=0.8 and P(Y<=1)=0.2+0.1=0.3

hence  P(X<=1 and Y <=1)=P(X<=1)*P(Y<=1)

therefore Yes

c)

P(X+Y=0)=P(0,0)=0.1

d)P(X+Y<=1)=P(0,0)+P(0,1)+P(1,0) =0.21

p(x,y) y x 0 1 2 3 4 0 0.1 0.05 0.025 0.025 0.3 1 0.06 0.03 0.015 0.015 0.18 2 0.04 0.02 0.01 0.01 0.12

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