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- Losses follow a Pareto distribution with parameters alpha= 2, theta =1000 For a coverage with policy limit 2000 and after an inflation rate of 10%, The expected cost after inflation isA scientist, posted on an island in the pacific. One of the instruments have is a seismometer. Notice there is no high ground on the island, so keep an eye on it in case it warns of a larger earthquake which could trigger a Tsunami. Remember that some people think earthquakes are distributed Poisson, and on the 2400 the hour (100 day anniversary) you get your 29th earthquake sensed on the seismometer. Using the Poisson assumption, A) calculate the maximum likelihood estimate of the waiting time between quakes.(hint the rate ? is 1/mean waiting time) B) Calculate an approximate 90% confidence interval for the waiting time using the normal approximationAll arriving patients at a clinic first sign in with a single receptionist. Suppose that arrivals occuraccording to a Poisson process with rate 5 patients per hour. If we know that the sign-upprocessing times are normally distributed with a mean of 10 minutes and a standard deviation of1 minute, what is the expected number of patients in the system in its statistical equilibrium?
- Let expected return and standard deviation of a risky portfolio are 10% and 18% respectively. Risk-free rate is 5%. If your risk-aversion coefficient is 2.5 (with this value your utility is maximised), what will be the expected return and standard deviation of the complete portfolio at the point where your indifference curve is tangent to the CAL?The occurrence of floods in a county follows a Poisson distribution with a return period of 20 years. The damage in each flood is log-normally distributed with a mean of $2 million and a coefficient of variation of 25%. Assume that damage in any one flood is statistically independent of the damage in any other flood. (a) What is the probability of more than two floods occurring in the county during the next 10 years? (b) What is the probability that damage in the next flood will exceed $3 million? (c) What is the probability than none of the floods that could occur in the next 10 years will cause damage exceeding $3 million?For the p parameter of Bernoulli (p) distribution; Find the maximum likelihood estimator.
- MLE: Maximum likelihood estimationIn a particular city, the population proportion π supports an increase in the minimum wage. For a random sample of size 2, let Y = number who support an increase. 1. Assuming π = 0.50, specify the probabilities for the possible values y for Y and find the distribution’s mean and standard deviation. 2. Suppose you observe y = 1 and do not know π. Find and sketch the likelihood function. Using the plotted likelihood function, explain why the ML estimate πˆ = 0.50.The probabilities of various numbers of failures in a mechanical test are as follows: Pr[0 failures] = 0.21, Pr[l failure] = 0.43, Pr[2 failures] = 0.28, Pr[3 failures] = 0.08, Pr[more than 3 failures] = 0. (a)Show this probability function as a graph with number of failures in x-axis and probability in y-axis. (b)Sketch a graph of the corresponding cumulative distribution function. Hint: cumulative probability in y-axis (c)What is the expected number of failures, E, or the mathematical expectation of the number of failures?
- Consider an auto insurance portfolio where the number of accidents follows a Poisson distribution with parameter lamda = 1000. Suppose the damage sizes for separate accidents are i.i.d. (independent identically distributed) r.v.'s having an exponential distribution with a mean of $2500. Each policy involves a deductible of $500. Let N1 be the number of accidents that result in claims, and N2 be the number of accidents that do not result in claims. Answer the following questions 1-5. 1. Are N1, N2 dependent? 2. What is the name of the distributions of N1, N2? Compound Poisson or Marked Poisson? 3. What is the mean and variance of N1? 4. What is the mean and variance of N2? 5. Which of the number is close to the probability that N1 will not be larger than 790? (Hint: to what distribution is the Poisson distribution with a parameter of lamda close for "large" lamda's?Assuming a loss frequency of a loss occurring once every 2 years and that the average value of losses that occur might be $40,000. Based on the above examples the expected loss isStanding eye heights of women normally distributed with a mean of 1516 mm and a standard deviation of 63 mm. What % of women have a standing eye height between 1420 mm and 1560 mm, and what is the probability that a group of 20 women has an average standing eye height that is less than 1500 mm? Even though the sample size is less than 30, why can the Central Limit Theorem still be applied?
