Question

(a) With a variable life insurance policy, the rate of return on the investment (the death benefit) varies from year to year. A study of these variable return rates was published in a certain journal. A transformed ratio of the return rates (x) for two consecutive years was shown to have a normal distribution, with µ = 2.6 and σ = 0.4. Compute the probabilities below, providing a sketch of the relevant curve, area, and x values for each case.
1. P(2.1 < x < 2.9) =

2. P(x > 2.5) =

3. P(x < 2.6) =

4. P(2.3 < x < 3.1) =

(b) Suppose x is a normally distributed random variable with µ = 51 and σ = 7. Find a value x0 of the random variable x that satisfies the following equations or statements, providing a sketch of the relevant curve, area, and x values for each case.
1. P(x ≥ x0) = 0.7513

2. P(x ≤ x0) = 0.025

3. P(x ≤ x0) = 0.95

4. P(50 ≤ x ≤ x0) = 0.8919

Answer 1