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Chapter 4: Q60E (page 240)

Question: Network forensic analysis. A network forensic analyst is responsible for identifying worms, viruses, and infected nodes in the computer network. A new methodology for finding patterns in data that signify infections was investigated in IEEE Transactions on Information Forensics and Security (May, 2013). The method uses multiple filters to check strings of information. For this exercise, consider a data string of length 4 bytes (positions). Where each byte is either a 0 or a 1 (e.g., 0010). Also, consider two possible strings, named S1and S2. In a simple single-filter system, the probability that and differ in any one of the bytes is .5. Derive a formula for the probability that the two strings differ on exactly x of the 4 bytes. Do you recognize this probability distribution?

Short Answer

Expert verified

The x is approximately a binomial random variable, and the x follows a binomial distribution with n=4 and p=0.5.

Step by step solution

01

Given information

The data string of length is 4 bytes.

The number of possible two strings isS1andS2.

The probability that S1andS2and differ in any one of the bytes is 0.5.

02

State the characteristics of the Binomial experiment

The characteristics of the Binomial experiment are as follows:

  • 路The experiments consist of n trials
  • 路Each trial consists of two outcomes only, either success or failure.
  • 路The probability of success is represented as 鈥減鈥�, and the probability of failure is defined as 鈥渜鈥�.
  • 路 The trials are independent to each other.
03

Check whether the random variable x represents a binomial random variable or not

The data string of length is 4 bytes. So, the number of trials is n=4.

The two possible outcomes are:

  • 路Success, S= the string that matches with the byte.
  • 路 Failure, F= the line does not match with the byte.

  • For each trial,

    The probability of success is,
    P = P (S)
    =0.5

    The probability of failure is,
    q=1-p=1-0.5=0.5



    The probability of success is the same for all the trials.

    Here, the trials are closely independent because one byte does not affect the matching of the other bytes.

    The random variable x is a data string of the length of 4 bytes or trials.

    Hence, the x is approximately a binomial random variable,and the x follows a binomial distribution with n=4 and p=0.5.

    i.e; x - B ( 4,0.5 )

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Most popular questions from this chapter

Industrial filling process. The characteristics of an industrialfilling process in which an expensive liquid is injectedinto a container were investigated in the Journal of QualityTechnology(July 1999). The quantity injected per containeris approximately normally distributed with mean 10

units and standard deviation .2 units. Each unit of fill costs\(20 per unit. If a container contains less than 10 units (i.e.,is underfilled), it must be reprocessed at a cost of \)10. A properly filled container sells for $230.

a. Find the probability that a container is underfilled. Notunderfilled.

b. A container is initially underfilled and must be reprocessed.Upon refilling, it contains 10.60 units. Howmuch profit will the company make on thiscontainer?

c. The operations manager adjusts the mean of the fillingprocess upward to 10.60 units in order to makethe probability of underfilling approximately zero.

Under these conditions, what is the expected profit percontainer?

If x is a binomial random variable, compute for each of the following cases:

  1. n = 4, x = 2, p = .2
  2. n = 3, x = 0, p = .7
  3. n = 5, x = 3, p = .1
  4. n = 3, x = 1, p = .9
  5. n = 3, x = 1, p = .3
  6. n = 4, x = 2, p = .6

Blood diamonds. According to Global Research News (March 4, 2014), one-fourth of all rough diamonds produced in the world are blood diamonds, i.e., diamonds mined to finance war or an insurgency. (See Exercise 3.81, p. 200.) In a random sample of 700 rough diamonds purchased by a diamond buyer, let x be the number that are blood diamonds.

a. Find the mean of x.

b. Find the standard deviation of x.

c. Find the z-score for the value x = 200.

d. Find the approximate probability that the number of the 700 rough diamonds that are blood diamonds is less than or equal to 200.

Checkout lanes at a supermarket. A team of consultants working for a large national supermarket chain based in the New York metropolitan area developed a statistical model for predicting the annual sales of potential new store locations. Part of their analysis involved identifying variables that influence store sales, such as the size of the store (in square feet), the size of the surrounding population, and the number of checkout lanes. They surveyed 52 supermarkets in a particular region of the country and constructed the relative frequency distribution shown below to describe the number of checkout lanes per store, x.

a. Why do the relative frequencies in the table represent the approximate probabilities of a randomly selected supermarket having x number of checkout lanes?

b. FindE(x) and interpret its value in the context of the problem.

c. Find the standard deviation of x.

d. According to Chebyshev鈥檚 Rule (Chapter 2, p. 106), what percentage of supermarkets would be expected to fall within渭卤蟽? within渭卤2蟽?

e. What is the actual number of supermarkets that fall within? ? Compare your answers with those of part d. Are the answers consistent?

Assume that xis a binomial random variable with n = 100

and p = 5. Use the normal probability distribution to approximate

the following probabilities:

a.P(x鈮�48)

b.P(50鈮�x鈮�65)

c.P(x鈮�70)

d.P(55鈮�x鈮�58)

e.P(x=62)

f.P(x鈮�49orx鈮�72)
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