Final answer:
The next train's waiting time is uniformly distributed from 1 to 41 minutes. It's a continuous distribution, with a mean of 21 minutes and a standard deviation of approximately 11.5470. Probabilities for specific time ranges were calculated, as well as the 34th percentile and a conditional probability.
Step-by-step explanation:
The time for the next train to come follows a Uniform distribution with density function f(x) = 1/40 for x between 1 and 41 minutes.
- The type of distribution is Uniform distribution.
- It is a continuous distribution.
- The mean of this distribution is the midpoint between 1 and 41 minutes, which is (1+41)/2 = 21 minutes.
- The standard deviation of this distribution is calculated using the formula σ = √((b-a)^2/12) which in this case is σ = √((41-1)^2/12) = √(1600/12) = √(133.3333) = 11.5470.
- The probability that the time will be at most 30 minutes is ∫ f(x) dx from x=1 to x=30 which gives (30-1)/40 = 0.7250.
- The probability that the time will be between 9 and 16 minutes is (16-9)/40 = 0.1750.
- The 34th percentile of this distribution is calculated by the formula x = a + P(b-a), where P is the percentile in decimal form; thus, x = 1 + 0.34(40) = 14.6 minutes.
- For the conditional probability that the time is more than 28 minutes given that it is at least 2 minutes, we use the formula P(A|B) = P(A and B) / P(B). Therefore, the probability is (41-28)/(41-2) = 13/39 = 0.3333.