A store sells a particular nonperishable product. The demand for the product each day is 1 item with probability 1/6, 2 items with probability 3/6, and 3 items with probability 2/6. Assume that the daily demands are independent and identically distributed. The store uses an (s, S) = (2, 4) policy to manage its inventory: each evening if the remaining stock is less than or equal to s items, the store orders enough to bring the total stock up to S items next morning. These items reach the store before the beginning of the following day. Assume that any demand is lost when the item is out of stock. Assume that each item sells at $150, the variable cost per item is cu = $50, the fixed cost for each order is cf = $100, and the holding cost is $5 for each item held overnight (for accounting). Let Xn be the inventory level at the end of day n. Then X = {Xn : n = 0,1,2,...,} is a discrete time Markov chain. Find the state space S and the transition matrix of the DTMC.

The table shows population statistics for the ages of best actor and best supporting actor winners at an awards ceremony. the distributions of the ages are approximately bell-shaped. compare the z-scores for the actors in the following situation. best actor best supporting actor muequals42.0 muequals49.0 sigmaequals7.3 sigmaequals15 in a particular year, the best actor was 59 years old and the best supporting actor was 45 years old. determine the z-scores for each. best actor: z equals best supporting actor: z equals (round to two decimal places as needed.) interpret the z-scores. the best actor was (more than 2 standard deviations above more than 1 standard deviation above less than 1 standard deviation above less than 2 standard deviations below) the mean, which (is not, is) unusual. the best supporting actor was (less than 1 standard deviation below more than 1 standard deviation above more than 2 standard deviations below more than 1 standard deviation below) the mean, which (is is not) unusual.

In the figure below, ∠dec ≅ ∠dce, ∠b ≅ ∠f, and segment df is congruent to segment bd. point c is the point of intersection between segment ag and segment bd, while point e is the point of intersection between segment ag and segment df. the figure shows a polygon comprised of three triangles, abc, dec, and gfe. prove δabc ≅ δgfe.

Find the length of the normal and the angle it makes with the positive x-axis. (see pictured below)