Need this answered as soon as possible, any help / explanations is appreciated.

Question

Need this answered as soon as possible, any help / explanations is appreciated. Thanks.

Suppose that the lifetime of parts from supplier 1 has a pdf 2y, 0 y 1, and the lifetime of parts from supplier 2 has a pdf ???, 0-y 1. Šuppose that 1/3 of the parts come from supplier 1 and 2/3 of the parts come from supplier 2 (a) If a part is chosen at random and found to have a lifetime of y, determine the probab?lity that it came from supplier (b) For what values of y is it more likely that the part came from supplier 1 than from supplier 2?

Explanation / Answer

P( Supplier 1 | Y= y ) = P( Supplier 1 and Y= y)/P(Y=y)

P( Supplier 1 and Y= y) = P(Y=y | Supplier 1) P(Supplier 1) = 2yX 1/3 = (2y)/3

Similarly,

P( Supplier 2 and Y= y) = P(Y=y | Supplier 2) P(Supplier 2) = 3y^2 X 2/3 = (2y^2)

P(Y=y) = P( Supplier 1 and Y= y) + P( Supplier 2 and Y= y) = (2y^2) + (2y)/3

P( Supplier 1 | Y= y ) = ( (2y)/3) / ((2y^2) + (2y)/3) = 1 / (1 + 1.5y)

To find y such that

P( Supplier 1 | Y= y ) > P( Supplier 2 | Y= y )

(2y)/3 > (2y^2)

y< 3/4

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