Irwin$$s sells a particular model of fan, with most of the sales being made in the summer months. Irwin$$s makes a one$-$time purchase of the fans prior to each summer season at a cost of $$40$ each and sells each fan for $$60.$ Any fans unsold at the end of the summer season are marked down to $$29$ and sold in a special fall sale. Virtually all marked$-$down fans are sold. The following is the number of sales of fans during the past $10$ summers: $30,50,30,60,10,40,30,30,20,40.$ Based on the observed $10$ values of the prior demand, construct an empirical probability distribution of summer demand and determine the optimal number

of fans for Irwin$$s to buy based on the empirical distribution.