Macroeconomics 1

Suppose that PrArt is decreasing for 0 0, s(x) = (1 -x/w)" (05xsu) Give simple formulae for (i) (ii) er (Note finite limiting age), (iii) 10P70; (iv) 40 5935 A certain group of lives now aged 60 experience mortality according to a(55) males ultimate with addition to the force of mortality. The addition is 0.0005 at age 60, increasing linearly to 0.0025 at age 80, at which level the addition remains constant. Find the probability that a life aged exactly 60 dies within 20 years.An actuarial student has said that the following three distributions are the same: (i) the chi square distribution with 2 degrees of freedom (ii) the exponential distribution with mean 12 (ii) the gamma distribution with a = 1 and 1 =K. State with reasons whether the student is correct. [2] The number of telephone calls per hour on a working day received at an insurance office follows a Poisson distribution with mean 2.5. (i) Calculate the probability that more than 7 telephone calls are received on a working day between 9am and 11am. [1] (ii) Calculate the probability that, if the office opens at 8am, there are no telephone calls received until after 9am. [2] [Total 3] The random variable x has an exponential distribution with parameter 1 . Use the moment generating function to determine an expression for E[X" ]. [3] A random variable X has probability density function: f(x) = 2x52 (5+ x13' *20 (i) Determine an expression for the distribution function, F(x) . [1] (ii) Calculate two simulated observations from the distribution using the random numbers 0.656 and 0.285 selected from the U(0, 1) distribution. [3] [Total 4] The random variable X has a beta distribution with parameters a =1 and =4. (i) State the value of E(X) . [1] (ii) Determine the median of X . [3] (iii) Hence comment on the shape of this distribution. [1] [Total 51