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) If x and Y are independent with p_x(0) = 0.5, p_x(1) = 0.3, p_x(2) = 0.2, and p_y(0) = 0.6, p_y(1) = 0.2, p_y(2) = 0.05, p_y(4) = 0.1, display the joint pmf of (X, Y) in a joint probability table. (b) Compute P(X lessthanorequalto 1 and Y lessthanorequalto 1) from the joint probability table. P(X lessthanorequalto 1 and Y lessthanorequalto 1) = Does P(X lessthanorequalto 1 and Y lessthanorequalto 1) equal the product P(X lessthanorequalto 1), P(Y lessthanorequalto 1)? Yes No (c) What is p(X + Y = 0) (the probability of no violations)? P(X + Y = 0) = (d) Compute P(X + Y lessthanorequalto 1). P(X + Y lessthanorequalto 1) =

Explanation / Answer

from property of independence P(X=x,Y=y) =P(X=x)*P(Y=y) joint probability table can be given as :

b)P(x<=1 and y<=1)= 0.3+0.1+0.18+0.06= 0.64

Yes it is equal s P(X<=1)=0.5+0.3=0.8 and P(Y<=1)=0.6+0.2=0.8 hence P(X<=1)*P(PY<=1)=0.8*0.8=0.64

c)P(X+Y=0)=P(X=0; Y=0)=0.3

d)P(X+Y<=1)=P(X=0;Y=0)+P(X=1;Y=0)+P(X=0;Y=1) =0.3+0.1+0.18=0.58

please revert for any clarification required

y P(x,y) 0 1 2 3 4 total x 0 0.3 0.1 0.025 0.025 0.05 0.5 1 0.18 0.06 0.015 0.015 0.03 0.3 2 0.12 0.04 0.01 0.01 0.02 0.2 total 0.6 0.2 0.05 0.05 0.1 1

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