The sample space of a random experiment is {a, b, c, d, e, f}, and each outcome is equally likely. A random variable is defined as follows: outcome…

1. The sample space of a random experiment is {a, b, c, d,

e, f}, and each outcome is equally likely. A random variable is

defined as follows:

outcome a b c d e f

x 0 0 1.5 1.5 2 3

Determine the probability mass function of a. Use the

probability mass function to determine the following

probabilities:

(a) P(X = 1.5) (b) P(0.5 < X < 2.7)

(c) P(X > 3) (d) P(0 ≤ X < 2)

(e) P(X = 0 or X = 2)

2. verify that the following functions are probability mass functions, and determine the requested

probabilities.

f (x)= 2x + 1/25

x=0, 1, 2, 3, 4

(a) P(X = 4) (b) P(X ≤ 1)

(c) P(2 ≤ X < 4) (d) P(X > −10)

3. In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is

classified as pass or fail. Assume that the probability that a wafer passes the test is 0.8 and that

wafers are independent. Determine the probability mass function of the number of wafers from a

lot that pass the test.

4. A disk drive manufacturer sells storage devices with capacities of one terabyte, 500

gigabytes, and 100 gigabytes with probabilities 0.5, 0.3, and 0.2, respectively. The revenues

associated with the sales in that year are estimated to be $50 million, $25 million, and $10

million, respectively. Let X denote the revenue of storage devices during that year. Determine

the probability mass function of X.

5. Determine the cumulative distribution function for the random variable

f (x)= 2x + 1/25

x=0, 1, 2, 3, 4

(a) P(X = 4) (b) P(X ≤ 1)

(c) P(2 ≤ X < 4) (d) P(X > −10)

6. F(x) = 0 x<10

0.25 10

≤ x< 30

0.75 30 ≤ x < 50

1 50 ≤ x

(a) P(X ≤ 50) (b) P(X ≤ 40)

(c) P(40 ≤ X ≤ 60) (d) P(X < 0)

(e) P(0 ≤ X < 10) (f) P(−10 < X < 10)

7. Determine the mean and variance of the random variable

outcome a b c d e f

x 0 0 1.5 1.5 2 3

(a) P(X = 1.5) (b) P(0.5 < X < 2.7)

(c) P(X > 3) (d) P(0 ≤ X < 2)

(e) P(X = 0 or X = 2)

8. Determine the mean and variance of the random variable

f (x)= 2x + 1/25

x=0, 1, 2, 3, 4

(a) P(X = 4) (b) P(X ≤ 1)

(c) P(2 ≤ X < 4) (d) P(X > −10)

9. Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99.

Determine the mean and variance of X.

10. Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine

the mean, variance, and standard deviation of the random variable Y = 5X and compare to the

corresponding results for X.

11. The random variable X has a binomial distribution with n = 10 and p = 0.01. Determine the

following probabilities.

(a) P(X = 5) (b) P(X ≤ 2)

(c) P(X ≥ 9) (d) P(3 ≤ X < 5)

12. An article in Information Security Technical Report[“Malicious Software—Past, Present and

Future” (2004, Vol. 9,pp. 6-18)] provided the following data on the top 10 malicious software

instances for 2002. The clear leader in the number of registered incidences for the year 2002

was the Internet worm “Klez,” and it is still one of the most widespread threats. This virus was

first detected on 26 October 2001, and it has held the top spot among malicious software for the

longest period in the history of virology.

The 10 most widespread malicious programs for 2002

Place†Name†•†Instances

1 IWorm.

Klez 61.22%

2 IWorm.

Lentin 20.52%

3 IWorm.

Tanatos 2.09%

4 IWorm.

BadtransII 1.31%

5 Macro.Word97.Thus 1.19%

6 IWorm.

Hybris 0.60%

7 IWorm.

Bridex 0.32%

8 IWorm.

Magistr 0.30%

9 Win95.CIH 0.27%

10 IWorm.

Sircam 0.24%

(Source: Kaspersky Labs). Suppose that 20 malicious software instances are reported.

Assume that the malicious sources can be assumed to be independent.

(a) What is the probability that at least one instance is “Klez?”

(b) What is the probability that three or more instances are “Klez?”

(c) What are the mean and standard deviation of the number of

“Klez” instances among the 20 reported?

13. Suppose that the random variable X has a geometric distribution with a mean of 2.5.

Determine the following probabilities:

(a) P(X = 1) (b) P(X = 4) (c) P(X = 5)

(d) P(X ≤ 3) (e) P(X > 3)

14. In a clinical study, volunteers are tested for a gene that has been found to increase the risk

for a disease. The probability that a person carries the gene is 0.1.

(a) What is the probability that four or more people need to be

tested to detect two with the gene?

(b) What is the expected number of people to test to detect two

with the gene?

15. A player of a video game is confronted with a series of opponents and has an 80%

probability of defeating each one. Success with any opponent is independent of previous

encounters. Until defeated, the player continues to contest opponents.

(a) What is the probability mass function of the number of

opponents contested in a game?

(b) What is the probability that a player defeats at least two

opponents in a game?

(c) What is the expected number of opponents contested in a game?

(d) What is the probability that a player contests four or more

opponents in a game?

(e) What is the expected number of game plays until a player

contests four or more opponents?

16. Consider the time to recharge the flash in cellphone

cameras as in Example 32.

Assume

that the probability that a camera passes the test is 0.8 and the cameras perform independently.

Determine the following:

(a) Probability that the second failure occurs on the tenth camera

tested.

(b) Probability that the second failure occurs in tests of four or

fewer cameras.

(b) Expected number of cameras tested to obtain the third

failure.

17. Suppose that X has a hypergeometric distribution with N = 100, n = 4, and K = 20.

Determine the following:

(a) P(X = 1) (b) P(X = 6)

(c) P(X = 4) (d) Mean and variance of X

18. A state runs a lottery in which six numbers are randomly selected from 40 without

replacement. A player chooses six numbers before the state’s sample is selected.

(a) What is the probability that the six numbers chosen by a player match all six numbers in the

state’s sample?

(b) What is the probability that five of the six numbers chosen

by a player appear in the state’s sample?

(c) What is the probability that four of the six numbers chosen

by a player appear in the state’s sample?

(d) If a player enters one lottery each week, what is the expected

number of weeks until a player matches all six numbers in

the state’s sample?

19. Suppose that X has a Poisson distribution with a mean of 0.4. Determine the following

probabilities:

(a) P(X = 0) (b) P(X ≤ 2)

(c) P(X = 4) (d) P(X = 8)

20. The number of telephone calls that arrive at a phone exchange is often modeled as a

Poisson random variable. Assume that on the average there are 10 calls per hour.

(a) What is the probability that there are exactly 5 calls in one hour?

(b) What is the probability that there are 3 or fewer calls in one hour?

(c) What is the probability that there are exactly 15 calls in two hours?

(d) What is the probability that there are exactly 5 calls in 30 minutes?

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