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Advanced Probability Exercises - Chapter 7
1. Classification of Random Variables (Discrete vs. Continuous)
A city transportation analyst records two variables over a month:
- X: the number of times a passenger takes the subway per day.
- Y: the total daily distance (in km) that the passenger travels on public transport.
a. Identify the possible values of X and Y.
b. Determine if each variable is discrete or continuous and justify.
c. Are the values of X and Y countable? Finite?
2. Validity of a Probability Distribution
Consider the following table for a variable Z: | z | -2 | 0 | 1 | 3 | 5 | |---|----|---|---|---|---|
| P(z) | 0.2 | 0.4 | 0.1 | p | 0.1 |
a. Determine the value of p that makes this a valid distribution.
b. Is the expected value of Z positive or negative? Explain without calculation.
3. Building a Discrete Probability Distribution from Real Data
A tech company surveyed 10,000 households about the number of smart devices they own:
| Devices Owned | 0 | 1 | 2 | 3 | 4+ |
|---------------|---|---|---|---|----|
| Households | 250 | 2200 | 3900 | 3000 | 650 |
a. Construct the probability distribution for the random variable D: number of smart devices owned.
b. What is the probability that a household owns at least two devices?
4. Compound Probability: Delivery Times
A delivery service has the following distribution of delivery days:
| Days | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Advanced Probability Exercises - Chapter 7
|------|---|---|---|---|---|---|---|
| P(X) | 0.01 | 0.04 | 0.10 | 0.30 | 0.35 | 0.15 | 0.05 |
a. What is the probability that a package is delivered within the promised 3-5 day window?
b. If a customer expects a delay past day 4, what is the conditional probability that they receive it on day 6 or later?
5. Expected Value and Variance with Transformation
A game has the following outcomes: | Score X | -5 | 0 | 3 | 10 | |---------|----|---|---|----|
| P(X) | 0.1 | 0.4 | 0.3 | 0.2 | Let Y = 2X - 1. a. Compute E(X), Var(X).
b. Compute E(Y), Var(Y) both directly and using transformation rules.
6. Tree Diagram & Combined Probability
You flip three biased coins where P(H) = 0.6 and P(T) = 0.4 for each. a. Draw the probability tree.
b. Find the probability of getting exactly two heads.
c. What is the probability of getting at least one tail?
7. Applying Discrete Distributions in Business
A store sells packs of snacks with this distribution:
| X = snacks | 1 | 2 | 3 | 4 | 5 |
|------------|---|---|---|---|---|
| P(X) | 0.05 | 0.15 | 0.35 | 0.25 | 0.20 |
Each snack costs $0.80; they sell each pack for $3.00.
Advanced Probability Exercises - Chapter 7
a. Calculate the expected profit per pack.
b. Find the variance of profit.
8. Decision Making via Expected Value
You're offered two investment options: - Option A: Guaranteed $500.
- Option B: 40% chance to earn $400, 30% chance to earn $900, 30% chance to earn $100.
a. Based on expected value, which should you choose?
b. What if Option A is reduced to $450?
9. Real-World Event Modeling
A university tracks the number of library visits per semester:
| Visits X | 0 | 5 | 10 | 20 | 30 | 40 | 50 |
|----------|---|----|----|----|----|----|----|
| P(X) | 0.10 | 0.20 | 0.25 | 0.20 | 0.15 | 0.07 | 0.03 |
a. Compute the mean and standard deviation of visits.
b. If each visit lasts 2 hours, find the expected total hours per semester.
10. Advanced Variance Problem with Money
An arcade charges $0.50 per game. Distribution:
| X = games | 1 | 2 | 3 | 4 | 5 |
|-----------|---|---|---|---|---|
| P(X) | 0.1 | 0.2 | 0.4 | 0.2 | 0.1 |
a. Find the mean and variance of games played.
b. Compute expected revenue per visitor and its variance.