Uniform Problem A subway train on the Red Line arrives every 8 minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution. Let X = the length of time a commuter must wait for the train to arrive, in minutes 0  X  8 X ~ U(0, 8)

Uniform Problem Graph the probability distribution. a = 0 b = 8

Uniform Problem f(x) =  =  =

Uniform Problem f(x) =  =  =

Uniform Problem P(X < 1) = (1 – 0)(1/8) = 1/8 Find the probability that the commuter waits less than one minute. P(X < 1) = (1 – 0)(1/8) = 1/8

Uniform Problem P(2 < X < 5) = (5 – 2)(1/8) = 3/8 Find the probability that the commuter waits between two and five minutes. P(2 < X < 5) = (5 – 2)(1/8) = 3/8

Uniform Problem Let k = the 90th %ile 0.90 = (k – 0)(1/8) k = 7.2 min. Find the 90th percentile. (90% of the wait times are less than this value.) Let k = the 90th %ile 0.90 = (k – 0)(1/8) k = 7.2 min.

Uniform Problem 0.60 = (8 – k)(1/8) 4.8 = 8 – k k = 3.2 60% of the time the commuter waits more than how long for the train? 0.60 = (8 – k)(1/8) 4.8 = 8 – k k = 3.2

Uniform Problem P(X > 5 | X > 3) = (8 – 5)(1/5)=3/5 CONDITIONAL: Find the probability that the commuter waits more than five minutes when he/she has already waited more than three minutes. P(X > 5 | X > 3) = (8 – 5)(1/5)=3/5