Chapter 4: Discrete Random Variables
Chapter 4 Homework
Section 4.1 Homework
Exercises
Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in [link].
| x | P(x) |
|---|---|
| 3 | 0.05 |
| 4 | 0.40 |
| 5 | 0.30 |
| 6 | 0.15 |
| 7 | 0.10 |
- In words, define the random variable X.
- What does it mean that the values zero, one, and two are not included for x in the PDF?
Section 4.2 Homework
Exercises
A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 a piece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150.
- What are you interested in here?
- In words, define the random variable X.
- List the values that X may take on.
- Construct a PDF.
- If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?
Solution
- I am interested in the average profit or loss.
- Let X = the return from the raffle
- Win($150) or Lose ($0)
-
Net Gain Probability $150 [latex]\frac{4}{100}[/latex] $0 [latex]\frac{0}{100}[/latex] - [latex]150(\frac{4}{100})+0(\frac{99}{100})−20=− \$14[/latex]
Exercises
A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.
- If the card is a face card, and the coin lands on Heads, you win $6
- If the card is a face card, and the coin lands on Tails, you win $2
- If the card is not a face card, you lose $2, no matter what the coin shows.
- Find the expected value for this game (expected net gain or loss).
- Explain what your calculations indicate about your long-term average profits and losses on this game.
- Should you play this game to win money?
Solution
The variable of interest is X, or the gain or loss, in dollars.
The face cards jack, queen, and king. There are [latex](3)(4) = 12[/latex] face cards and [latex]52 – 12 = 40[/latex] cards that are not face cards.
We first need to construct the probability distribution for X. We use the card and coin events to determine the probability for each outcome, but we use the monetary value of X to determine the expected value.
| Card Event | X net gain/loss | P(X) |
|---|---|---|
| Face Card and Heads | 6 | [latex]\left(\frac{12}{52}\right)\left(\frac{1}{2}\right)=\left(\frac{6}{52}\right)[/latex] |
| Face Card and Tails | 2 | [latex]\left(\frac{12}{52}\right)\left(\frac{1}{2}\right)=\left(\frac{6}{52}\right)[/latex] |
| (Not Face Card) and (H or T) | –2 | [latex]\left(\frac{40}{52}\right)\left(1\right)=\left(\frac{40}{52}\right)[/latex] |
- [latex]\text{Expected value}=\left(6\right)\left(\frac{6}{52}\right)+\left(2\right)\left(\frac{6}{52}\right)+\left(-2\right)\left(\frac{40}{52}\right)=–\frac{32}{52}= – \$0.62[/latex]
- If you play this game repeatedly, over a long string of games, you would expect to lose 62 cents per game, on average.
- You should not play this game to win money because the expec