1. Camer Pharmaceuticals is testing a new product in the market1. The demand for the new product is estimated to be Normally distributed with a mean 2,000,000 and standard deviation 250,000. The demand is estimated to grow at a rate of 4% per year. The R&D costs are estimated to be between $500 millions of dollars and $800 millions of dollars with a most likely value of $700 millions of dollars. Clinical trial costs are estimated to be between $135 millions of dollars and $160 millions of dollars with a most likely value of $150 millions of dollars. There are competitors in the market, and Camer Pharmaceuticals estimates that their market share in the first year will be any number between 6% and 10%, with each number being equally likely. The company estimates that their market share will grow by 20% each year. A monthly prescription is estimated to generate a revenue of $240. The variable costs are estimated to be $30. Develop a simulation model that calculates the net present value (NPV) of this project over 3 years assuming a discount rate of 10%. Run the simulation for 1000 iterations and answer the following questions.

a. What is the distribution of the NPV (mean and standard deviation)?

b. What are the first quartile, median, and third quartile of NPV?

c. What is the probability that NPV over 3 years will not be positive?

d. What NPV are we likely to observe with a probability of at least 0.9?

e. What cumulative net profit in the third year are we likely to observe with a probability of at least 0.9?

f. What is the 95% confidence interval for the mean NPV? Interpret the resulting confidence interval.

g. What is the number of iterations needed if we want to estimate the NPV within $4,000,000?

h. Interpret the sensitivity chart.

Suppose we have a distribution that is unimodal, bell-shaped, and roughly symmetric. What is the name of the result that tells us that we should expect approximately 95% of our data to fall within two standard deviations from the mean? O Law of Averages O Empirical Rule Central Limit Theorem O Law of Large Numbers QUESTION 24 Decide whether the random variable is discrete or continuous. The random variable X equals the number of cars that run a red light during a specified amount of time. O Discrete O Continuous O Cannot determine we would need to know the duration of time.Pointwise Asymptotic Normality of the Empirical CDF 2.0/3 points (graded) Let X1, . .., Xn ~ P for some distribution P, and let F denote its cdf. Let F, denote the empirical cdf. Then it holds for every t c IR that (d) Vn (Fn (t) - F(t)) N (0, 02) n +00 for any fixed t and for some asymptotic variance o2. What theorem implies the above convergence statement? Central limit theorem. Law of large numbers. Glivenko-Cantelli theorem. Which of the following is o? ? Note that of is dependent on t. OF(t) O1-F(t) OVF(t) ( 1 -F(+) ) OF(t) (1 - F(t) ) What is the asymptotic variance o' of F (0) in terms of the values of the cdf F? (Enter F(x) for F' (@) for any numerical value 23. Let X1, X2, . . . be i.i.d. U[0, 1] random variables. Dene Xn = n1 :1 X1. (a) Show, by the Weak Lavr of Large Numbers, that X converges in distribution to 1/2. (b) Show, by the Weak Law of Large Numbers, that 711 ESQ-X2? converges in distribution to 1 / 3. (c) Show, by the Central Limit Theorem, that HQ(n 1/2) con- verges in distribution to a N (0, 1 / 12) random variable. ((1) The Central Limit Theorem asserts that 731/2 Z?=I(Xf 1/3) converges in distribution to a random variable Z. Specify the distribution of Z. Which of the following is not true? Choose the correct answer below. O A. A z-score is an area under the normal curve. O B. If values are converted to standard z-scores, then procedures for working with all normal distributions are the same as those for the standard normal distribution. O C. The area in any normal distribution bounded by some score x is the same as the area bounded by the equivalent z-score in the standard normal distribution O D. A z-score is a conversion that standardizes any value from a normal distribution to a standard normal distribution