Probability Distributions: Solved Practice Problems

1. Exponential Distribution: Tax Office Service Time

The service time (in minutes) at the local government tax office is exponentially distributed with λ = 0.025. (15 points)

• a) What are the mean and standard deviation?
  μ = σ = 1/λ = 40 minutes
• b) Find the probability that the service time will be greater than 40 minutes.
  e^{-0.025 × 40} = 0.3678
• c) Find the probability that the service time will be between 20 and 35 minutes.
  (1 - e^{-0.025 × 35}) - (1 - e^{-0.025 × 20}) = 0.583 – 0.393 = 0.19

2. Uniform Distribution: Barcelona Subway Waiting Time

The time a person waits for the Barcelona subway is uniformly distributed from 0 to 6 minutes. (15 points)

• a) What is the probability of each outcome?
  p = 1/(b - a) = 1/6
• b) What are the mean and standard deviation of waiting times?
  μ = (0 + 6)/2 = 3; σ = √[(b - a)²/12] = 1.73
• c) What is the probability that a person will wait between 3 and 5 minutes?
  Area = base × height = 2 × 1/6 = 1/3

3. Normal Distribution: Light Bulb Working Life

A company produces light bulbs with an average working life following a normal distribution N(500, 50). (15 points)

• a) 2% of bulbs are defective due to low working life. What is the maximum number of hours to be considered defective?
  Z = -2.05; -2.05 = (X – 500) / 50; X = 397.5 hours
• b) 5% of bulbs are defective due to long working life. What is the minimum number of hours to be considered defective?
  Z = 1.65; 1.65 = (X – 500) / 50; X = 582.5 hours

4. Normal Probability Plot Analysis

Think about the chart below in relation to the normal distribution. What is it? What does it suggest? (15 points)

This is the normal probability plot, a graphical technique for normality testing to assess whether a data set is approximately normally distributed. In this case, it indicates a right skew, meaning there is a long tail to the right of the distribution function.

5. Binomial vs. Normal Distribution

How do the binomial and the normal distribution functions differ? How do they relate? (20 points)

The binomial is a discrete distribution, whereas the normal is a continuous distribution. However, the binomial can be approximated to the normal when np > 5 and n(1 - p) > 5 (where n is sample size and p is the probability of success). If the binomial is distributed as B(n, p) and these conditions hold, then B(n, p) ≈ N(np, √np(1-p)).

6. Normal Distribution: Test Results

The results of a partial test follow a normal distribution with a mean of 65.5 and a standard deviation of 15. (15 points)

• a) The top 10% of the class avoids the final exam. What is the minimum grade required?
  Z = 1.29; 1.29 = (X – 65.5) / 15; X = 84.85
• b) The bottom 25% must complete extra work. What is the maximum result to avoid this?
  Z = -0.675; -0.675 = (X – 65.5) / 15; X = 55.375