# Discrete Random Variable

Exercise 1

Complete the PDF and answer the questions.

|X |P(X = x) |X(P(X = x) |

|0 |0.3 | |

|1 |0.2 | |

|2 | | |

|3 |0.4 | |

a. Find the probability that X = 2.

b. Find the expected value.

Exercise 2

Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6. a. What are you ultimately interested in here (the value of the roll or the money you win)? b. In words, define the Random Variable X.

c. List the values that X may take on.

d. Construct a PDF.

e. Over the long run of playing this game, what are your expected average winnings per game? f. Based on numerical values, should you take the deal? g. Explain your decision in (f) in complete sentences.

Exercise 3

A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars.

a. Construct a PDF for each investment.

b. Find the expected value for each investment.

c. Which is the safest investment? Why do you think so? d. Which is the riskiest investment? Why do you think so? e. Which investment has the highest expected return, on average? Exercise 4

A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase 4 tickets. The prize is 2 passes to a Broadway show, worth a total of $150.

a. What are you interested in here?

b. In words, define the Random Variable X.

c. List the values that X may take on.

d. Construct a PDF.

e. If this fund-raiser is repeated often and you always purchase 4 tickets, what would be your expected average winnings per game? Exercise 5

Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children |X |P(X = x) |X(P(X = x) |

|0 |0.10 | |

|1 |0.20 | |

|2 |0.30 | |

|3 | | |

|4 |0.10 | |

|5 |0.05 | |

|6 (or more) |0.05 | |

a. Find the probability that a married adult has 3 children. b. In words, what does the expected value in this example represent? c. Find the expected value.

d. Is it more likely that a married adult will have 2 – 3 children or 4 – 6 children? How do you know?

Exercise 6

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given below.

|X |P(X = x) |

|3 |0.05 |

|4 |0.40 |

|5 |0.30 |

|6 |0.15 |

|7 |0.10 |

a. In words, define the Random Variable X.

b. What does it mean that the values 0, 1, and 2 are not included for X on the PDF? c. On average, how many...

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