Question

The commuter trains on the blue and green lines for the regional transit authority (rta) in cleveland, oh, have a waiting time during peak rush hour periods of ten minutes ("2012 annual report," 2012). a.) state the random variable. b.) find the probability of waiting between two and five minutes. c.) find the probability of waiting between seven and ten minutes. d.) find the probability of waiting eight minutes exactly.

Answer 1

The random variable (X) refers to the wait time for a train, and it follows a uniform distribution between 0 to 8 minutes (X~U(0,8)). Probabilities of waiting within specific intervals can be calculated using the ratio of the interval to the total range, but the exact time probabilities are 0 for continuous distributions.

Step-by-step explanation:

The random variable (X) in these situations represents the time a commuter must wait for a train to arrive. Since the trains arrive every eight minutes and the waiting time is uniformly distributed, we can denote this as X~U(0,8), where 'U' stands for Uniform distribution and the range of 0 to 8 represents the minimum and maximum possible waiting times.

• Probability of waiting between two and five minutes: To calculate this, we look at the span of time (5-2) and divide by the total span of time (8-0).
• Probability of waiting between seven and ten minutes: This is outside the range of the uniform distribution, as the maximum waiting time is 8 minutes. Thus, the probability is 0.
• Probability of waiting exactly eight minutes: For continuous distributions like the uniform distribution, the probability of waiting for an exact amount of time is technically 0, as there are an infinite number of points within any range.