ASSIGNMENT Chapter 4Statistical MethodsNAME:M&S 1844.4 (6 points) Type of Random Variable. Identify the following random variables as discrete or continuous:a. The amount of flu vaccine in a syringeb. The heart rate ( number of beats per minute ) of an American malec. The time it takes a student to complete an examination d. The barometric pressure at a given locatione. The number of registered voters who vote in a national electionf. You score on the SATANSWERSa. The amount of flu vaccine in a syringe is measured on an interval, so this is a continuous random variable. b. The heart rate (number of beats per minute) of an American male is countable, starting at whatever number of beats per minute is necessary for survival up to the maximum of which the heart is capable. That is, if m is the minimum number of beats necessary for survival, x can take on the values (m, m + 1, m + 2, ...) and is a discrete random variable. c. The time necessary to complete an exam is continuous as it can take on any value 0 ≤ x ≤ L, where L = limit imposed by instructor (if any). d. Barometric (atmospheric) pressure can take on any value within physical constraints, so it is a continuous random variable. e. The number of registered voters who vote in a national election is countable and is therefore discrete. f. An SAT score can take on only a countable number of outcomes, so it is discrete.M&S 187-188, 2234.15 (3 points) A discrete random variable x can assume five possible values: 2,3,5,8, and 10. Its probability distribution is shown here:xp(x) 2 3 5 8 10 .15 .10 .25 .25a. What is p (5) ?b. What is the probability that x equals 2 or 10?c. What is P (x  8)?ANSWERS a. .25 b. .40 c. .754.22 (2 points) controlling the water hyacinth. Entomologists are continually searching for new biological agents to control the water hyacinth, one of the world’s worst aquatic weeds. an insect that naturally feeds on the water hyacinth is the delphacid. Female delphacids lay anywhere form one to four eggs onto a water hyacinth blade. The Annals of the Entomological Society of American (Jan. 2005) published a study of the life cycle of a South American delphacid species. The following table gives the percentages of water hyacinth blades that have one, two, three, and four delphacod eggs: One Egg Two Eggs Three eggs Four eggsPercentage ofBlades 40 54 2 4 Source: Sosa, A.J., et al. “Life history of Megamelus scutellaris with decription of immature stages.” Annals of the Entomological Society of America, Vol. 98, No.1, 2005 (adaptes from Table 1).a. One of the water hyacinth blades in the study is randomly selected, and x, the number of delphacid eggs on the blade, is observed. Give the probability distribution of x. b. What is the probability that the blade has at least three delphacid eggs?ANSWERSQuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.4.134 (5 points) Reliability of a bridge network. In the journal Networks (May 1995), a team of Chinese university professors investigated the reliability of several capacitates-flow networks. One network examined in the article and illustrated here is a bridge network with arcs a1, a2, a3, a4, a5, and a6. The probability distribution of the capacity x foreach of the six arcs is provided in the following table: Arc Capacity x p (x) Arc Capacity x p (x) a1 3 .60 2 .25 1 .10 0 .05 a4 1 .90 0 .10 a2 2 .60 1 .30 0 .10 a5 1 .90 0 .10 a3 1 .90 0 .10 a6 2 .70 1 .25 0 .05Source: Lin, J., et al. “On reliability evaluation of capacitated-flow network in terms of minimal pathsets.” Networks, Vol. 25, No. 3, May 1995, p.135 (Table 1) a1 a2       Source ---------- ----------- Sink node node        a5 a6 a3 a4a. Verify that the properties of a discrete probability distribution are satisfied for each are capacity distribution.b. Find the probability that the capacity for arc a1 will exceed 1.c. Repeat part b for each of the remaining five arcs.d. Compute  and  for each arc, and interpret the results.e. One path from the source node to the sink node is through arcs a1 and a2. Find the probability that the system maintains a capacity of more than 1 through the a1-a2 path. (Assume that the are capacities are independent.)f. Another path from the source node to the sink node is through a1, a3, and a6. Find the probability that the system maintains a capacity of 1 through the a1-a3-a6 Path.ANSWERS QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.M&S 195-1964.37 (3 points) Consider the probability distributions shown here: xp (x) 0 1 2 .3 .4 .3 yp (y) 0 1 2 .1 .8 .1 a. Use your intuition to find the mean for each distribution. How did you arrive atyour choice?b. Which distribution appears to be more variable? Why?c. Calculate  and 2 for each distribution. Compare these answers