Question

The engineer responsible for the maintenance of a dam is in charge of keeping the water level in it within certain limits. There is a water level known as level 0, from which certain security measures are applied. For the dam control purposes, the level of the water is considered each thousand cubic hectometers (hm3). When the monitoring system is put in place, the water level is one thousand hm3 above level 0. Each day the engineer monitors the water level and applies the following policy: if the water level in the dam is grater than or equal to 3 thousand hm3 above level 0, he lets out enough water to leave the level of the water at 2 thousand hm3, if the water level is less than 3 thousand but more than (or equal to) 2 thousand hm3 above level 0 then he lets out 1 thousand hm3 of water, if the water level is less than 2 thousand hm3 above level 0 at the time of inspection then he does nothing. Keep in mind that only multiples of thousands are considered for the water level. The probability distribution of the amount of rain over the dam on any given day is the following :

P(amount of rain = 0 hm3) = 3/8
P(amount of rain = 1 thousand hm3) = 1/8
P(amount of rain = 2 thousand hm3) = 4/8
Express the situation of the water level control of the dam as a Markov chain, by defining the variable as xt = water level in the dam before the engineer lets the water out according to the policy at period t.
IN YOUR DOCUMENT ANSWER THE FOLLOWING:
(a) Write the one-step transition matrix for this problema (hint: note that state 0 is not posible) (17 pts)
(b) If the monitoring system is put in place on Tuesday, find the probability that the water level in the dam will be at 2 thousand hm3 on Friday of the same week. (8 pts)
IN THE SPACE PROVIDED WRITE THE PROBABILITY FOR PART (b) TRUNCATED TO FOUR DECIMALS:

Answer 1

The probability that the

water level

in the dam will be at 2 thousand hm3 on Friday of the same week is approximately 0.0234.

Step-by-step explanation:

To express the situation of the

water level control

of the

dam

as a

Markov chain

, we define the variable xt as the water level in the dam before the engineer applies the policy at period t. The possible states for xt are 1 thousand hm3, 2 thousand hm3, and 3 thousand hm3 above level 0.

(a) The

one-step transition matrix

for this problem can be calculated using the given probabilities of the

amount of rain

over the dam. Let's denote the transition matrix as P, where P[i][j] represents the

probability

of transitioning from state i to state j.

Since state 0 is not possible, we only consider the transitions from states 1, 2, and 3 thousand hm3 above level 0. The transition probabilities are as follows:

(b) To find the probability that the water level in the dam will be at 2 thousand hm3 on

Friday

of the same week, we need to calculate the probability of transitioning from the initial state (1 thousand hm3 above level 0) to the desired state (2 thousand hm3 above level 0) in three steps. We can use matrix multiplication to calculate this probability.

Let's denote the initial state as X0 and the desired state as X3. The probability of transitioning from X0 to X3 in three steps can be calculated as:

P(X0 to X3 in 3 steps) = P[1][2] * P[2][3] * P[3][2]