Question

1. Consider the following joint relative frequency table f(x,y). y= x= A B 10 0.200 0.100 20 0.075 0.150 30 0.100 0.025 с 0.050 0.200 0.100 The table entries represent relative frequencies of items of type A, B and C sold in boxes containing 10, 20 or 30 items each. For example, the relative frequency 0.200 in the first column and first row shows that 20% of all items sold were items of type A in boxes containing 10 items each. Similarly, 15% of all items sold were of type B in boxes containing 20 items each, etc. a. Does the relative frequency distribution satisfy the properties of a (joint) probability distribution. Explain. b. Regardless of your finding in (a) assume that the relative frequencies constitute a joint probability distribution. Find the probability that a randomly chosen item was of type C and was sold in a box containing 30 items. C. Find the probability that a randomly chosen item was of type B regardless of the box size (quantity in the box) (a marginal probability). d. Find the probability that a randomly chosen item was sold in a box containing 30 items regardless of item type (marginal probability). e. Find the marginal distribution g(x) for X, the number of items in a box, where X=10, 20, 30. (You may consider expanding the above table appropriately). f. Find the expected value or mean for the number sold [v = E(X)). (You may consider expanding the above table appropriately). g. Find the standard deviation 6 for the number of items sold. (You may consider expanding the above table appropriately).

Answer 1

a. In the relative frequency distribution , all the entries are less than 1 and greater than

0 and moreover, the sum of the entries is equal to 1. Thus it is a joint probability

distribution.

b. P(Box C and number of items is 30) = 0.1

c. P(Box B) = 0.1+0.15+0.025 = 0.275

d. P(Number of item = 30 ) = 0.1+0.025+0.1 = 0.225

e.

X	P(X=x)
10	0.35
20	0.425
30	0.225

f. E(X) = 0.35*10+0.425*20+0.225*30 = 18.75

g. E(X^2) = 0.35*100+0.425*400+0.225*900 = 407.5

Var( X) =407.5 - 18.75^2 = 55.9375

Standard Deviation = $\sqrt$ 55.9375 = 7.48

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