A box contains 100 pieces of jewelry Each piece is either a

A box contains 100 pieces of jewelry. Each piece is either a necklace or a ring with

equal probabilities. The price of each ring is a continuous uniform random variable in the interval

(0,9) and the price of each necklace is also a continuous uniform random variable in the interval

(0, 3).

(a) Let X be the price of a randomly chosen piece of jewelry from the box. Find the expectation

and the variance of X.

(b) If S is the total price of all pieces in the box, use the Central Limit Theorem to estimate the

probability that S is less than 300 + 10square root 6

P(ring) = 0.5

P(necklace) = 0.5

x price of ring is uniform in (0,3) and y is uniform in (0,9)

E(x) = Mean of uniform = 1.5

Var(x) = 9/12 = 0.75

b) Total prices of all pieces = 50x+50y=z

E(Z) = 50(1.5)+50(4.5)

= 300

Var(z) = 2500 (varx+vary)

= 2500(0.75 + 6.67)

= 18550

std error = 136.198

Prob for S < 300+10sq rt 6 = 1-1/6 = 5/6