EX 3.23 repurchase unsold computer

• Scenario Overview:

  • The computer store purchases 3 computers at $500 each.
  • Selling price per computer: $1000.
  • The manufacturer will repurchase unsold computers at $200 each.

• Definition of Random Variable:

  • Let $X$ represent the number of computers sold.

• Probability Mass Function:

  • The probabilities for the number of computers sold are:
    • $p(0)=0.1$
    • $p(1)=0.2$
    • $p(2)=0.3$
    • $p(3)=0.4$

• Profit Function:

  • The profit associated with selling $X$ units is defined as:
  $$h(X)=\text{Revenue}-\text{Cost}$$
  • The profit can be expressed as:
  $$h(X)=1000X+200(3-X)-1500$$
  • Simplifying this gives:
  $$h(X)=1000X+600-200X-1500=800X-900$$

• Conclusion:

  • The profit function $h(X)=800X-900$ captures the financial outcome based on the number of computers sold.
  • Understanding the distribution of $X$ will allow for calculations of expected profit and decision-making for the store.
  • The expected profit is then

$$\begin{array}{rl}E[h(X)]& =h(0)\cdot p(0)+h(1)\cdot p(1)+h(2)\cdot p(2)+h(3)\cdot p(3)\\ & =(-900)(0.1)+(-100)(0.2)+(700)(0.3)+(1500)(0.4)\\ & =\$700\end{array}$$