Problem:

Corporations have to select among many projects that are under consideration by the management. Their primary instrument for evaluating and selecting among the available projects is the benefit-cost analysis. In this analysis, both the annual benefits and the annual costs deriving from a project are estimated in several different categories. Then the total benefit is divided by the total cost to produce a benefit-cost ratio. This ratio is then used by corporations to compare numerous projects under consideration. A benefit-cost ratio greater than 1.0 indicates that the benefits are greater than the costs, and the higher a project's benefit-cost ratio, the more likely it is to be selected over projects with lower ratios.

Currently, the JET Corporation is evaluating two dam project constructions, one in southwest Georgia (Dam #1) and the other in North Carolina (Dam #2). The company has identified six areas of benefits: improved navigation, hydroelectric power, fish and wildlife, recreation, flood control, and the commercial development of the area. Furthermore, there are three estimates available for each type benefit - a minimum possible value, a most likely value (i.e., a mode or peak), and a maximum possible value. For the costs, two categories associated with a construction project of this type have been identified: the total capital cost, annualized over 30 years (at a rate specified by the creditors and the government), and the annual operations and maintenance costs. These benefits and costs estimations for both dam projects (in millions of dollars) are as follows:

# Define benefits for Dam 1

B1 = np.array([1.1, 2, 2.8])

B2 = np.array([8, 12, 14.9])

B3 = np.array([1.4, 1.4, 2.2])

B4 = np.array([6.5, 9.8, 14.6])

B5 = np.array([1.7, 2.4, 3.6])

B6 = np.array([0, 1.6, 2.4])

# Define costs for Dam 1

C1 = np.array([13.2, 14.2, 19.1])

C2 = np.array([3.5, 4.9, 7.4])

# Define benefits for Dam 2

B1_d2 = np.array([2.1, 3, 4.8])

B2_d2 = np.array([8.7, 12.2, 13.6])

B3_d2 = np.array([2.3, 3, 3])

B4_d2 = np.array([5.9, 8.7, 15])

B5_d2 = np.array([0, 3.4, 3.4])

B6_d2 = np.array([0, 1.2, 1.8])

# Define costs for Dam 2

C1_d2 = np.array([12.8, 15.8, 20.1])

C2_d2 = np.array([3.8, 5.7, 8])

Part 1 of question:

(ii) Construct both a tabular and a graphical frequency distribution for "a1" and "a2" separately (a tabular and a graphical distribution for "a1", and a tabular and a graphical distribution for "a2"- a total of 4 distributions). In your report, include only the graphical distributions and comment on the shape of each distribution.

get me both the tabular frequency distribution codes in R for both a1 and a2. as well as the graphical frequency distribution codes for both a1 and a2.

based on this code:

# Define benefits and costs for Dam 1 B1 <- c(1.1, 2, 2.8) B2 <- c(8, 12, 14.9) B3 <- c(1.4, 1.4, 2.2) B4 <- c(6.5, 9.8, 14.6) B5 <- c(1.7, 2.4, 3.6) B6 <- c(0, 1.6, 2.4) C1 <- c(13.2, 14.2, 19.1) C2 <- c(3.5, 4.9, 7.4) # Define benefits and costs for Dam 2 B1_d2 <- c(2.1, 3, 4.8) B2_d2 <- c(8.7, 12.2, 13.6) B3_d2 <- c(2.3, 3, 3) B4_d2 <- c(5.9, 8.7, 15) B5_d2 <- c(0, 3.4, 3.4) B6_d2 <- c(0, 1.2, 1.8) C1_d2 <- c(12.8, 15.8, 20.1) C2_d2 <- c(3.8, 5.7, 8) # Function to perform simulation and calculate benefit-cost ratio sim_bcr <- function(B1, B2, B3, B4, B5, B6, C1, C2) { benefit <- c(rbeta(1e4, B1[2], B1[3]), rbeta(1e4, B2[2], B2[3]), rbeta(1e4, B3[2], B3[3]), rbeta(1e4, B4[2], B4[3]), rbeta(1e4, B5[2], B5[3]), rbeta(1e4, B6[2], B6[3])) cost <- c(rbeta(1e4, C1[2], C1[3]), rbeta(1e4, C2[2], C2[3])) bcr <- sum(benefit) / sum(cost) return(bcr) } # Perform simulation for Dam 1 set.seed(123) a1 <- replicate(1e4, sim_bcr(B1, B2, B3, B4, B5, B6, C1, C2)) # Perform simulation for Dam 2 set.seed(456) a2 <- replicate(1e4, sim_bcr(B1_d2, B2_d2, B3_d2, B4_d2, B5_d2, B6_d2, C1_d2, C2_d2)) # Create frequency distribution plot for Dam 1 hist(a1, breaks = 50, col = "blue", main = "Frequency Distribution of Benefit-Cost Ratio for Dam 1") # Create frequency distribution plot for Dam 2 hist(a2, breaks = 50, col = "green", main = "Frequency Distribution of Benefit-Cost Ratio for Dam 2")