The operational state of a machine in a factory over a period of 625 days (rough

Question

The operational state of a machine in a factory over a period of 625 days (roughly 1.7 years) is represented by a sequence of independent and identically distributed Bernoulli random variables, {X_i, i = 1, 2,..., 625}. If the machine is operational on day i, then the corresponding random variable X_i = 1. and if the machine is out of order on day i, then X_i = 0. Further, it is known that P(X_i = 1) = 0.8. A repair firm contracts with the factory to fix the machine as soon as it fails. The cost to the repair firm is $100 per visit to fix the machine. The repair firm for this company has the following long-term contract: The repair firm charges a one-time contract fee of $10,000. In addition, the following clauses hold: The first 105 visits are free (thus, the factory pays only the contract fee of $10,000). If the number of times the repair firm visits the factory exceeds 105, The factory pays the repair firm an additional sum of $10,000 (thus, in this case, including the contract fee, the factory pays a total of $10,000 + $10,000 = $20,000). Determine an expression for the random variable corresponding to the Total Profit (i.e., Total money received - Total cost) made by the repair firm (denote this by P). In other words, determine an expression for P in terms of {X_i, i = 1, 2,..., 625}. Next, using a central limit theorem approximation, determine a numerical value for the expected profit made by the repair firm (You may use the CDF table).

Explanation / Answer

X = 625 - Xi (i is from 1 to 625)

P = 10000 + I(X > 105)*1000 - 100X

From the distribution of Xi, we can get

E[Xi] = 0.8 and Var[Xi] = 0.16

Thus,

(X - 125) / sqrt(625*0.16) ~ N(0,1)

(X - 125) / 10 ~ N(0,1)

Now we have,

E[I(X > 105)] = P(X > 105)

= 1 - (105 - 125 / 10)

= 1 - (-2)

This probability we can find by using EXCEL.

syntax,

=NORSMDIST(z)

where z = -2

= 1 - 0.0228

= 0.9772

E[P] = 10000 + E[ I(X>105) ]*10000 - E[X]*100

= 10000 + 9772 - 12500

= 7272

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