Expected number of offers until one clears 9000

Question

A seller takes the first offer worth at least $9000 for his car. Offers arrive independently, each exponential with mean $5000. Counting the accepted offer, the expected number of offers he receives equals $e^a$ for some rational $a$ . Find $a$ .

Short answer

$a=9/5$ . Each offer clears the bar with probability $\mathbb{P}(B\ge9000)=e^{-9/5}$ , so the number of offers is geometric with that success probability and mean $e^{9/5}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Random Variables and Probability Mass Functions
Independence
Expectation, Mean, and Variance

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