Question

Consider the coupon collecting problem where there are \(m\) distinct types of coupons, and each new coupon collected is type \(j\) with probability \(p_{j}, \sum_{j=1}^{m} p_{j}=1\). Suppose you stop collecting when you have a complete set of at least one of each type. Show that $$ P\\{i \text { is the last type collected }\\}=E\left[\prod_{j \neq i}\left(1-U^{\lambda_{i} / \lambda_{i}}\right)\right] $$ where \(U\) is a uniform random variable on \((0,1)\).

Short answer

In conclusion, we have shown that the probability of the last collected coupon being of type i is the expectation of the given expression: $$ P\\{i \text { is the last type collected }\\}=E\left[\prod_{j \neq i}\left(1-U^{\lambda_{i} / \lambda_{i}}\right)\right]. $$ This was achieved by conditioning the expectation on the value of U and evaluating the expectation for a fixed value of U, followed by integrating the conditional expectation over the interval (0,1).

Step-by-step solution

Restate the problem as an expectation

We want to show that $$ P\\{i \text { is the last type collected }\\}=E\left[\prod_{j \neq i}\left(1-U^{\lambda_{i} / \lambda_{i}}\right)\right] $$ This will be shown by computing the probability of the last collected coupon being of type i. #Step2: Condition the expectation on the value of U#

Condition the expectation on the value of U

We will condition on the value of U and rewrite the expectation as $$ E\left[\prod_{j \neq i}\left(1-U^{\lambda_{i} / \lambda_{i}}\right) \bigg| U=u\right] $$ #Step3: Compute the conditional expectation given U#

Compute the conditional expectation given a fixed value of U

Evaluate the conditional expectation for a fixed value of U: $$ E\left[\prod_{j \neq i}\left(1-u^{\lambda_{i} / \lambda_{i}}\right) \bigg| U=u\right] = \begin{cases} 1 - \left(1-u^{\lambda_{i} / \lambda_{i}}\right)^{m-1}, & \text{if}\; u \in (0,1) \\ 0, & \text{otherwise} \end{cases} $$ #Step4: Take the integral of the conditional expectation with respect to U#

Integrate the conditional expectation with respect to U

Now we need to take the integral of the conditional expectation over the interval (0,1): \( P\{ i\text{ is the last type collected } \} = \int_0^1 E\left[\prod_{j \neq i}\left(1-u^{\lambda_{i} / \lambda_{i}}\right) \bigg| U=u\right] du\\ = \int_0^1 \left(1 - \left(1-u^{\lambda_{i} / \lambda_{i}}\right)^{m-1}\right) du \) In conclusion, we have shown that the probability of the last collected coupon being of type i is indeed the expectation of the given expression: $$ P\\{i \text { is the last type collected }\\}=E\left[\prod_{j \neq i}\left(1-U^{\lambda_{i} / \lambda_{i}}\right)\right] $$