Answer me please

Calculate the expectation and the variance of the present value of the financial loss for the insurance in Example 6.1.1, when the premium is determined by Principle I. Verify that the exponential premium (with a = 0.1) for the insurance in Ex- ample 6.1.1, modified so that the benefit amount is 10, is 3,45917. (Note that this is roughly 11.3 times as large as the exponential premium for a benefit amount of 1 found in Example 6.1.1.) Using the assumptions of Example 6.1.1, determine the annual premium that maximizes the expected utility of an insurer with initial wealth w = 10 and utility function u(x) = x - 0.01x', x 0) = 0.50. c. Repeat part (b) if the force of interest, 8, equals 0. If the force of mortality strictly increases with age, show that P(A,) > M, (0) [Hint: Show that P(A ) is a weighted average of p,(t), t > 0.] Following Example 6.2.1, derive a general expression for 2A, - (A, )2 (8a,)2 where u,(t) = p and 8 is the force of interest for t > 0.4) Forecasting (1 point) Your company is moving into a new area with 25% lower income on average. You've estimated that your product has an income elasticity of -0.5. By what percentage will your sales increase or decrease? 5) Double Take Beauty (Multiple Choice 1 point) Double Take Beauty is a spa in Provo, UT that offers single needle tattoos done by its founder, Sadie Flores. Suppose Sadie charges about $400 per tattoo on average. The marginal cost of each tattoo is about $100. She is getting better at tattooing and the tattoos are becoming more popular with customers, so she is considering raising the price to $500 per tattoo. What percentage of customers must be retained to ensure that the price increase is protable? Note: It's conventional when not calculating elasticities new value of Xold. value of): to measure percentage changes relative to thelr lmtlal value (%AX = old wk\" of x ). a. 18% b. 25% c. 66.67% (1. 75% e. 84% 6) Diminishing Returns (multiple choice 1 point) Diminishing returns to tractors indicates that: a. Having more tractors does not always lead to more output. 10. Having more tractors leads to increases in output only if there are corresponding increases in technology. c. Additional output may be produced with fewer tractors. d. Having more tractors leads to more output, but at a decreasing rate. 7) Economies of Scope (1 point) The function given by 021.02) = 2012 + 3027- 20102 + 201 12602 + 3000 Represents the minimum cost of producing given levels of output 1 (01) and output 2 (02) together. The minimum costs of producing 01 and Q; separately are given by 6021.0) = 203 + 201 + 3300 0(0, 02) = 30% 12602 + 3300 Are there economies of scope or diseconomies of scope in this case? Show your work to justify your answer. Ve = cosOv, + sin 0 v2 Vu =-sin 0 v1 + cos0 v2 After being produced, a neutrino travels freely with a definite energy and momentum, therefore its wavefunction is simply that of a plane-wave solution to the Schrodinger equation: [v(t)) = ei( px-Et)in| v(0)) Use relativistic kinematics to show that (px - Et) ~ -m c'x/2E for a particle of mass m traveling at nearly the speed of light. Then, apply the mixing formulas above to determine v(t) for a neutrino for which v(0)) = Ve . v(t)) will contain a term with vu; the square of its coefficient indicates the probability of measuring this neutrino as a muon-type after it has traveled a distance x. Show that this probability works out to be Peru = sin 2 20sin 2 (mz -m? )c'x 4Eh The second factor in this expression is responsible for the mechanism of neutrino oscillations