[43 points] Quality audit records are kept on the numbers of major and minor fai

Question

[43 points] Quality audit records are kept on the numbers of major and minor failures that occur to a certain type of circuit pack used during the burn-in period of large electronic switching devices. Let 1· X = the number of major failures Y-the number of minor failures Suppose that the random variables X and Y can be described, at least approximately, by the joint probability mass function given below. x.y) 0 1234 0 0.15 0.10 0.10 0.10 0.05 x 0.05 0.08 0.14 0.08 0.05 2 0.01 0.01 0.02 0.03 0.03

Explanation / Answer

Ans:

E(X)=0*0.5+1*0.4+2*0.1=0.6

E(Y)=0*0.21+1*0.19+2*0.26+3*0.21+4*0.13=1.86

std dev(X)=SQRT((0-0.6)^2*0.5+(1-0.6)^2*0.4+(2-0.6)^2*0.1)=0.66

std dev(Y)=SQRT((0-1.86)^2*0.21+(1-1.86)^2*0.19+(2-1.86)^2*0.26+(3-1.86)^2*0.21+(4-1.86)^2*0.13)=1.32

E(XY)=1*1*0.08+1*2*0.14+1*3*0.08+1*4*0.05+2*1*0.01+2*2*0.02+2*3*0.03+2*4*0.03=1.32

For X and Y to be independent:

P(X and Y)=P(x)*P(y)

P(X=0,Y=0)=0.15

P(x=0)*P(y=0)=0.5*0.21=0.105

Both are not equal,hence X and Y are not independent.

Cov(X,Y)=E(XY)-E(X)*E(Y)=1.32-0.6*1.86=0.204

Corr(X,Y)=0.204/(0.66*1.32)=0.234

y 0 1 2 3 4 Marginal(x) 0 0.15 0.1 0.1 0.1 0.05 0.5 x 1 0.05 0.08 0.14 0.08 0.05 0.4 2 0.01 0.01 0.02 0.03 0.03 0.1 Marginal(y) 0.21 0.19 0.26 0.21 0.13 1

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