Which of the following expresses the recursive relationship for ex using p_x and e_{x+1}?

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Multiple Choice

Which of the following expresses the recursive relationship for ex using p_x and e_{x+1}?

You’re dealing with the expected number of whole years a person aged x will live beyond age x (the curtate future lifetime). In one-year time steps, the only way to gain a full year is to survive the next year. If you survive from age x to x+1 (with probability p_x), you’ve completed one year and then still expect e_{x+1} more full years beyond age x+1. If you don’t survive the first year (with probability 1 − p_x), you don’t gain any additional full years. Put together, the expected number of future whole years is ex = p_x [1 + e_{x+1}]. This matches option where you multiply the chance of surviving the year by the one-year gain plus the remaining expectation after age x+1. The other forms would incorrectly introduce extra constants or reference later ages (like e_{x+2} or e_{x+3}) that don’t align with the one-year-step recursive decomposition for the curtate lifetime.

You’re dealing with the expected number of whole years a person aged x will live beyond age x (the curtate future lifetime). In one-year time steps, the only way to gain a full year is to survive the next year. If you survive from age x to x+1 (with probability p_x), you’ve completed one year and then still expect e_{x+1} more full years beyond age x+1. If you don’t survive the first year (with probability 1 − p_x), you don’t gain any additional full years.

Put together, the expected number of future whole years is ex = p_x [1 + e_{x+1}]. This matches option where you multiply the chance of surviving the year by the one-year gain plus the remaining expectation after age x+1.

The other forms would incorrectly introduce extra constants or reference later ages (like e_{x+2} or e_{x+3}) that don’t align with the one-year-step recursive decomposition for the curtate lifetime.

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