(a) The graph of the distribution of X is a bell-shaped normal curve, symmetric around the mean μ = 66.2 years, with a standard deviation σ = 4.4 years. The x-axis represents age at death in years, and the y-axis represents the probability density.

Key points to mark on the graph:

• Mean: 66.2 years (center of the curve).
• Standard deviation: Spread of 4.4 years (indicate ±1σ at 66.2 ± 4.4 = 61.8 and 70.6; ±2σ at 57.4 and 75.0; ±3σ at 53.0 and 79.4 for context).
• Specific decision points: Vertical lines or marks at 65 years, 70 years, and 75 years to highlight payment thresholds.

The curve peaks at 66.2 and tapers off symmetrically. The area to the right of 65 represents those receiving at least one payment, right of 70 for two or more, and between 70 and 75 for exactly two payments.

In practice, sketch this on graph paper provided in the exam, labeling axes and points clearly.

(b) The lifetimes follow X ~ N(66.2, 4.4²). Probabilities are calculated using z-scores: Z = (X – μ) / σ, then using standard normal tables (provided in the exam) to find cumulative probabilities.

(i) Percentage receiving at least one payment: P(X > 65)

Z = (65 – 66.2) / 4.4 = -1.2 / 4.4 = -0.2727

P(X > 65) = P(Z > -0.2727) = P(Z < 0.2727) ≈ 0.6075 (from normal table, P(Z < 0.27) ≈ 0.6064, P(Z < 0.28) ≈ 0.6103; interpolate to ≈0.6075)

Thus, 60.75%.

(ii) Percentage receiving two or more payments: P(X > 70)

Z = (70 – 66.2) / 4.4 = 3.8 / 4.4 ≈ 0.8636

P(Z < 0.8636) ≈ P(Z < 0.86) ≈ 0.8051, P(Z < 0.87) ≈ 0.8078; interpolate ≈0.8061

P(X > 70) = 1 – 0.8061 = 0.1939 or 19.39%.

(iii) Percentage receiving exactly two payments: Receives payments at 65 and 70, but not at 75, i.e., P(70 < X ≤ 75)

Z_70 ≈ 0.8636, P(Z ≤ 0.8636) ≈ 0.8061

Z_75 = (75 – 66.2) / 4.4 = 8.8 / 4.4 = 2.0

P(Z ≤ 2.0) = 0.9772

P(70 < X ≤ 75) = 0.9772 – 0.8061 = 0.1711 or 17.11%.