4. Bolts are manufactured for a certain purpose may be classified as acceptable

Question

4. Bolts are manufactured for a certain purpose may be classified as acceptable (suitable for the intended purpose), downgraded (unsuitable for the intended purpose but acceptable for a different purpose) or scrap (unsuitable for any purpose). In a lot of 500 bolts, let X be the number that are downgraded and let Y be the number that are scrap. Assume that the joint probability function is given in the following table. 0.06 0.03 0.01 0.00 0.00 1 0.06 0.08 0.04 0.02 0.00 2 0.04 0.05 0.12 0.06 0.03 3 0.00 0.03 0.07 0.09 0.06 4 0.00 0.00 0.02 0.06 0.07 a. Find the marginal probability mass function of px(x). (2 points) b. Find the marginal probability mass function of py(z). (2 points) c. Are X and Y independent? Explain. (3 points) d. Find the conditional probability mass function pyjx(v1). (8 points) e. Find the conditional expectation E(Y X-1). (2 points) f. Find the conditional probability mass function pxjy(|2). (2 points)

Explanation / Answer

a. px(x) = 0.10 ; x = 0

= 0.20; x = 1

= 0.30; x = 2

= 0.25; x = 3

= 0.15; x = 4

(b) py (y) = 0.16; y = 0

= 0.19; y = 1

= 0.26 ; y = 2

= 0.23 ; y = 3

= 0.16; y = 4

(c)X andY will be independent

if the joint probability distribution fX,Y (x,y) = fX(x) fy(y)

so for x = 0, and y = 0

fX,Y (x,y) = 0.06

fX(x) fy(y) = 0.16 * 0.10 = 0.016 0.06

so both x and y are not independet.

(d) pylx (y l1 ) =

pylx (y lx = 1 ) =

= 0.06/0.20 = 0.3 ; y = 0

= 0.08/0.20 = 0.4; y = 1

= 0.04/0.20 = 0.2; y = 2

= 0.02/0.20 = 0.1 ; y = 3

= 0 ; y = 4

(e) E(Y l X = 1) = 0.3 * 0 + 0.4 * 1 + 0.2 * 2 + 0.1 * 3 + 0 * 4 = 1.1

(f) pxly (xl2 ) =

pxly (xly = 2 ) = 0.01/0.26= 0.038 ; x = 0

= 0.04/0.26 = 0.154; x = 1

= 0.12/0.26 = 0.462; x = 2

= 0.07/0.26 = 0.269 ; x = 3

= 0.02/0.26 = 0.077 ; x= 4

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