- #1

mathmari

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A bookstore buys 15 copies of a book at a price of 20 euros each and offers them on sale for 30 euros each. The contract provides that the bookstore after a year can return the unsold copies of the book and receive 18 euros each.

Let the number of sold copies of this book in a year be determined by a random variable X with probability function $p_X(x)=\frac{x+2}{150}, \ x\in \{1,2, \ldots , 15\}$.

(a) Verify that $p_X$ is indeed a probability function.

(b) Determine the expected number of sold copies in a year. Write also the intermediate steps.

(c) Determine the bookstore's expected profit from the sale of this book if all unsold books are returned after one year. Write also the intermediate steps. I have done the following :

(a) Do we have to check that $p_X(\Omega)=1$ and $p_X\left (\cup A_i\right )=\sum p_X(A_i)$ ? :unsure:

(b) Is the expected number of sold copies equal to $E[X]=\sum_{i=1}^{15}x\cdot p_X(x)$ ? :unsure:

(c) The total expected profit is the sum of expected profits from each book, right? For one book teh expected profit is $(30-20)\cdot 0.50+(18-20)\cdot 0.50=4$, or isn't the probability to sell or not to sell equal to $\frac{1}{2}$ ? Is then the total profit $15\cdot 4=60$ ? :unsure: