1. 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)

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

3. Uniform Problem
f(x) =
 =
 =

4. Uniform Problem
f(x) =
 =
 =

5. 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

6. 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

7. 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.

8. 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

9. 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