The image shows a payment order for electricity to a certain energy sales company.

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-38.jpg?height=646&width=1380&top_left_y=1163&top_left_x=339)

Each month, the client provides the company with readings from a three-tariff meter installed in the apartment. The current month's readings are subtracted from the corresponding readings of the previous month to obtain the actual consumption for the month in each of the three tariff zones (peak, night, semi-peak). The consumption in each zone is then multiplied by the price per kilowatt-hour in that zone. By summing these amounts, the client gets the total payment for the month. In this example, the client will pay 660 rubles and 72 kopecks.

The company keeps track of electricity consumption and payments using the data provided by the client. The problem is that the company sometimes confuses the six numbers, rearranging them in any order, although it ensures that the current reading remains greater than the previous one. As a result, the company's calculation can be incorrect. If the company believes that the client owes more than they paid, it demands the difference.

Using the data from the illustrated receipt, find:

a) the maximum possible additional payment for March 2013 that the company may demand from the client;

b) the expected value of the difference between the amount the company will charge and the amount the client has paid.

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a) It is obvious that the amount the company will require will be the highest if the consumption at the highest tariff is the highest possible, and at the medium tariff - the highest possible from the remaining. The highest possible amount in this case is 4.03$\cdot$(1402 - 1214) + 3.39$\cdot$(1347 - 1270) + 1.01$\cdot$(1337 - 1298) = 1058.06 (rub.). The company will require an additional payment of 1058.06 - $660.72=397.34$ (rub.).

b) Let's solve the problem in general, assuming that the client provided six different numbers $a<b<c<d<e<f$. Consider one of the tariff zones (for example, Peak). For this zone, any two numbers from the given

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-39.jpg?height=69&width=2069&top_left_y=0&top_left_x=0)

![](https://cdn.mathpix.com/cropped/2024_05_08_22585010b2534c7ab9ebg-39.jpg?height=63&width=2056&top_left_y=66&top_left_x=1)
the average of fifteen numbers: $f-e, f-d, f-c, f-b, f-a, e-d, e-c, e-b, e-$ $a, d-c, d-b, d-a$,

$c-b, c-a, b-a$, that is, $\mathrm{E} X_{1}=1 / 15(5 f+3 e+d-c-3 b-5 a)$

Obviously, the random variables $X_{2}$ and $X_{3}$ for "Night Tariff" and "Semi-Peak Tariff" have the same expectation. The total payment amount $S$, calculated by the company, is $t_{1} X_{1}+t_{2} X_{2}+t_{3} X_{3}$, where $t_{1}, t_{2}$, and $t_{3}$ are the prices per kilowatt-hour for the respective tariffs. Therefore,

$\mathrm{E} S=t_{1} \mathrm{E} X_{1}+t_{2} \mathrm{E} X_{2}+t_{3} \mathrm{E} X_{3}=1 / 15(5 f+3 e+d-c-3 b-5 a)\left(t_{1}+t_{2}+t_{3}\right)$

In our case, $f=1402, e=1347, d=1337, c=1298, b=1270, a=1214 ; t_{1}+t_{2}+t_{3}=4.03+1.01+3.39=$ $8.43$. Thus,

$\mathrm{ES}=1 / 15(5 \cdot 1402+3 \cdot 1347+1337-1298-3 \cdot 1270-5 \cdot 1214) \cdot 8.43=1210 / 15 \cdot 8.43=680.02$

The expected value of the difference between the amount the sales company will require and the amount the client will pay is

$\mathrm{E} S-660.72=680.02-660.72=19.30$ (rub.)

## Answer

a) 397 rub. 34 kop.; b) 19 rub. 3 kop.