A business sells ice cream tubs for €2 each. The probability distribution for the number of tubs sold (X) is given. Calculate the expected revenue and profit for two cost functions: CA(x) = 0.3x + 400

Step 1: Define the variables and calculate the expected values of sales.
Let $X$ be the number of tubs sold.
The selling price per tub is €2.
The revenue function is $R(x) = 2x$.

The probability distribution for $X$ is given as:
$x_i$: 100, 200, 300, 400, 500
$P(x_i)$: 0.15, 0.20, 0.25, 0.28, 0.12

First, calculate the expected number of tubs sold, $E[X]$:

$$E[X] = \sum x_i P(x_i)$$ $$E[X] = (100 \times 0.15) + (200 \times 0.20) + (300 \times 0.25) + (400 \times 0.28) + (500 \times 0.12)$$ $$E[X] = 15 + 40 + 75 + 112 + 60$$ $$E[X] = 302 \text{ tubs}$$

Next, calculate the expected value of $X^2$, $E[X^2]$, as some cost functions depend on $x^2$:

$$E[X^2] = \sum x_i^2 P(x_i)$$ $$E[X^2] = (100^2 \times 0.15) + (200^2 \times 0.20) + (300^2 \times 0.25) + (400^2 \times 0.28) + (500^2 \times 0.12)$$ $$E[X^2] = (10000 \times 0.15) + (40000 \times 0.20) + (90000 \times 0.25) + (160000 \times 0.28) + (250000 \times 0.12)$$ $$E[X^2] = 1500 + 8000 + 22500 + 44800 + 30000$$ $$E[X^2] = 106800$$

The expected revenue is $E[R(x)] = E[2x] = 2E[X]$:

$$E[R(x)] = 2 \times 302 = 604 \text{ €}$$

Step 2: Calculate the expected profit for each procedure.
The profit $G(x)$ for each procedure is $G(x) = R(x) - C(x)$, where $C(x)$ is the cost function. We need to find $E[G(x)] = E[R(x)] - E[C(x)]$.

• Procedure A: Cost function: $C_A(x) = 0.3x + 400$ Expected cost:

  $$E[C_A(x)] = 0.3 E[X] + 400$$ $$E[C_A(x)] = (0.3 \times 302) + 400 = 90.6 + 400 = 490.6 \text{ €}$$

  Expected profit:

  $$E[G_A(x)] = E[R(x)] - E[C_A(x)] = 604 - 490.6 = 113.4 \text{ €}$$

• Procedure B: Cost function: $C_B(x) = 1x + 0.04\% x^2 = x + 0.0004x^2$ Expected cost:

  $$E[C_B(x)] = E[X] + 0.0004 E[X^2]$$ $$E[C_B(x)] = 302 + (0.0004 \times 106800) = 302 + 42.72 = 344.72 \text{ €}$$

  Expected profit:

  $$E[G_B(x)] = E[R(x)] - E[C_B(x)] = 604 - 344.72 = 259.28 \text{ €}$$

• Procedure C: Cost function: $C_C(x) = 0.5x + 0.06\% x^2 + 150 = 0.5x + 0.0006x^2 + 150$ Expected cost:

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