#Calculations please,

Using the assumption of a uniform distribution of deaths in each year of age and the Illustrative Life Table with interest at the effective annual rate of 6%, Calculate a. tire: IEism Ies b. Vari) for x = 20, 50, 80. [I Iint: Use {5.2.3}, {5.2.9}, and {4.431).} Using the values obtained in Exercise 5.1, calculate the standard deviation and the coefficient of variation, ow" u, of the following present-value random variables. a. Individual annuities issued at ages 20, 50, 80 with life incomes of 1,000 per year payable continuously. b. A group of 100 armuities, each issued at age 50, with life income of 1,000 per year payable continuously. Show that Varfe) can be expressed as 2 _ E (a: _ 2&1) _ HE! where 2:11, is based on the force of interest 23. Calculate Cov(5 , UT). If a deterministic (rate function) approach is adopted, (5.2.27) could be taken as the starting point for the development of a theory of continuous life an- nuities. For this, we would begin with da 5': [u(v}+la, 1 ravens r15, = [l or E y. a. Use the integrating factor exp{Ig[u(z) + 5M2} to solve the differential equation to obtain (5.2.3). b. Use the integrating factor e45\" to obtain a. = m f E's- a My) dy and give an interpretation of it in words. Assume that Mac + t) = p. and the force of interest is B for all t E 0. a. If Y = a T1, 0 E T, display the formula for the distribution function of Y. Y JaT OST <>