PSTAT 5A Practice Worksheet 4
Comprehensive Review: Discrete Random Variables and Distributions
Instructions and Overview
⏰ Time Allocation:
Quiz Review : 15 minutes
Section A (Warm-up): 15 minutes
Section B (Intermediate): 20 minutes
Optional Question: Do on your own
Total: 50 minutes
📝 Important Instructions:
Use the formulas provided for guidance
Round final answers to 4 decimal places unless otherwise specified
Identify the distribution type before calculating
Show your work for expected value and variance calculations
Use calculator as needed for factorials and combinations
📚 Key Formulas Reference:
General Random Variable Properties:
Expected Value: \(E[X] = \sum_{k} k \cdot P(X = k)\)
Variance: \(\text{Var}(X) = E[X^2] - (E[X])^2 = \sum_{k} k^2 \cdot P(X = k) - \mu^2\)
Standard Deviation: \(\sigma = \sqrt{\text{Var}(X)}\)
Discrete Distributions:
Bernoulli Distribution: \(X \sim \text{Bernoulli}(p)\)
- PMF: \(P(X = k) = p^k(1-p)^{1-k}\) for \(k \in \{0,1\}\)
- Mean: \(E[X] = p\)
- Variance: \(\text{Var}(X) = p(1-p)\)
Binomial Distribution: \(X \sim \text{Binomial}(n,p)\)
- PMF: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) for \(k = 0, 1, 2, ..., n\)
- Mean: \(E[X] = np\)
- Variance: \(\text{Var}(X) = np(1-p)\)
Geometric Distribution: \(X \sim \text{Geometric}(p)\)
- PMF: \(P(X = k) = (1-p)^{k-1} p\) for \(k = 1, 2, 3, ...\)
- Mean: \(E[X] = \frac{1}{p}\)
- Variance: \(\text{Var}(X) = \frac{1-p}{p^2}\)
Poisson Distribution: \(X \sim \text{Poisson}(\lambda)\)
- PMF: \(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\) for \(k = 0, 1, 2, ...\)
- Mean: \(E[X] = \lambda\)
- Variance: \(\text{Var}(X) = \lambda\)
Section A: Basic Concepts and Identification
⏱️ Estimated time: 15 minutes
Problem A1: Distribution Identification
For each scenario below, identify the appropriate discrete distribution and state the parameter(s). Do not calculate probabilities yet.
(a) A fair coin is flipped until the first head appears. Let X = number of flips needed.
(b) A quality control inspector tests 20 randomly selected items from a production line where 5% are defective. Let X = number of defective items found.
(c) A website receives visitors at an average rate of 3 per minute. Let X = number of visitors in a 2-minute period.
(d) A basketball player shoots one free throw with an 80% success rate. Let X = 1 if successful, 0 if unsuccessful.
(e) A student keeps taking a driving test until they pass. The probability of passing on any attempt is 0.7. Let X = number of attempts needed to pass.
Work Space:
Problem A2: Probability Mass Function
The random variable X has the following probability distribution:
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X=k) | 0.1 | 0.3 | 0.4 | a | 0.1 |
(a) Find the value of \(a\).
(b) Calculate \(P(X \leq 3)\).
(c) Calculate \(P(X > 2)\).
Work Space:
Section B: Expected Value and Variance
⏱️ Estimated time: 20 minutes
Problem B1: Manual Calculations
Using the probability distribution from Problem A2, calculate:
(a) The expected value \(E[X]\)
(b) The variance \(\text{Var}(X)\)
(c) The standard deviation \(\sigma\)
Calculation Strategy:
For expected value: \(E[X] = \sum k \cdot P(X = k)\)
For variance: First find \(E[X^2] = \sum k^2 \cdot P(X = k)\), then use \(\text{Var}(X) = E[X^2] - (E[X])^2\)
Show your work step by step!
Work Space:
Problem B2: Bernoulli and Binomial Applications
A manufacturing process produces items that are defective with probability 0.15.
(a) If you select one item randomly, what is the expected value and variance of X = number of defective items?
(b) If you select 25 items randomly, what is the expected number of defective items and the standard deviation?
Part (a) is a Bernoulli distribution. Part (b) is a Binomial distribution. Use the formulas from the reference box!
Work Space:
Optional Questions
Optional Problem : Conceptual Understanding
(a) Explain the key difference between a Binomial distribution and a Geometric distribution in terms of what they count.
(b) When would you use a Poisson distribution instead of a Binomial distribution?
(c) If \(X \sim \text{Binomial}(n, p)\), under what conditions would the variance be maximized?
Work Space: