I'm struggling hard with this problem since to be honest I have no clue from where to start. I'd love to share with you my efforts on solving but unfortunately I have none. I think I should use Poisson variable to solve it though. Here's the question: An album of figurines contains 150 figurines, which are sold in packs of 5 each. Using Approach
$$\sum_{k=1}^n \frac{1}{k} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} = \ln(n+0.5)+0.5772$$
Determine how many packages I need to buy on average to complete the album.
Thanks in advance!
Answer is 167.74
This is known as the coupon collector's problem. Let $N$ be a random variable representing the number of required buys to collect all $150$ figurines, and $n_i$ be the number of buys to collect the $i$th figurine after $i − 1$ figurines are collected for $i\in[1,150]$. The probability of getting a new figurine is given by $$p_i = \frac{150 − (i − 1)}{150}$$ and $n_i\sim {\tt Geometric}(1/p_i)$ whose mean is also $1/p_i$. Obviously, $$N=n_1+n_2+\cdots+n_{150}$$ Using the linearity of expectation we have: $$\begin{align} \mathsf{E}N&=\mathsf{E}n_1+\mathsf{E}n_2+\cdots+\mathsf{E}n_{150}\\ &=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_{150}}\\ &=\frac{150}{150}+\frac{150}{149}+\cdots+\frac{150}{1}\\ &=150\left(\frac{1}{150}+\frac{1}{149}+\cdots+\frac{1}{1}\right)\\ &=150(\ln(150+0.5)+0.5772)\\ &=838.6745 \end{align}$$
But since they are sold in packs of $5$, we need to divide it by $5$ which gives $167.7349$.