1. Let the pmf of ��� be defined by

���(���)

= ���

9

, ��� = 2,3,4.

a. Draw a bar graph for this pmf.

b. Draw a probability histogram for this pmf.

2. Let a random experiment be the cast of a pair of unbiased six-sided dice and let

��� equal the smaller of the outcomes if they are different and the common value if they are equal.

a. With reasonable assumptions, find the pmf of ���.

b. Draw a probability histogram.

c. Let ��� equal the range of the two outcomes (i.e., the absolute value of the difference of the largest and the smallest outcomes). Determine the pmf ���(���) of ��� for ��� = 0,1,2,3,4,5.

d. Draw the probability histogram for ���(���).

[���(���) = 13−2��� , ��� = 1,2,3,4,5,6; ���(0) = 6 , ���(���) = 12−2��� , ��� =

36 36

36

1,2,3,4,5]

3. In a lot of 100 light bulbs, there are 5 bad bulbs. An inspector inspects 10 bulbs selected at random. Find the probability of finding at least one defective bulb. HINT: First compute the probability of finding no defectives in the sample.

[0.416]

4. Let ��� be the number of accidents in a factory per week having pmf

1

���(���) = (��� + 1)(��� + 2) , ��� = 0,1,2, … .

Find  the  conditional  probability  of  ��� ≥ 4,  given  that ��� ≥ 1. HINT: Write

���(���) = 1 − 1 .

(���+1) (���+2)

[2/5]

5. Let the random variable ��� be the number of days that a certain patient needs to be in the hospital. Say ��� has the pmf

5 − ���

���(���) =

, ��� = 1,2,3,4 .

10

If the patient is to receive from an insurance company $200 for each of the first two days in hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization?

[$360]

6. In a particular lottery, 3,000,000 tickets are sold each week for 50 cents apiece. Out of the 3,000,000 tickets, 12,006 are drawn at random and without replacement and awarded prizes: twelve thousand $25 prizes, four $10,000 prizes, one $50,000 prize, and one $200,000 prize. If you purchased a single ticket each week, what is the expected value of this game to you?

[-91/300 or -30.33 cents]

7. A roulette wheel used in a United States casino has 38 slots of which 18 are red, 18 are black and 2 are green. A roulette wheel used in a French casino has 37 slots of which 18 are red, 18 are black and 1 is green. A ball is rolled around  the wheel and ends up in one of the slots with equal probability. Suppose that a player bets on red. If  a $1 bet is placed, the player wins $1 if the ball ends up  in a red slot (tge okater’s $1 bet is returned). If the ball ends up in a black or green slot, the player loses $1. Find the expected value of this game to the player in

a. The United States.

b. France.

[-1/19,-1/37]

8. Find the mean and variance for the following discrete distributions:

a. ���

(���)

= 1 , ��� = 5,10,15,20,25.

5

b. ���(���) = 4−��� , ��� = 1,2,3.

6

9. A measure of skewness is defined by

[15,50;5/3,5/9 ]

���[(��� − ���)³]

{���[(��� − ���)2]}3/2 =

���[(��� − ���)³]

{���2}3/2 =

���[(��� − ���)³]

���3 .

When a distribution is symmetrical about the mean, the skewness is equal to

zero. If the probability histogram has a longer “tail” to the right than to the left, the measure of skewness is positive, and we say that the distribution is skewed positively or to the right. If the probability histogram has a longer tail to the left than to the right, the measure of skewness is negative, and we say that the distribution is skewed negatively or to the left.  If the pmf of ��� is given by ���(���),

(i) depict the pmf as a probability histogram and find the values of (ii) the mean,

(iii) the standard deviation and (iv) skewness.

(  ) 2^{6−���}/64, ���  = 1,2,3,4,5,6

��� ��� = {

1/64, ��� = 7.

[127/64,√7359/64,1.6635]