Question

Today, the waves are crashing onto the beach every 5.8 seconds. The times from when a person arrives at the shoreline until a crashing wave is observed follows a Uniform distribution from 0 to 5.8 seconds. Round to 4 decimal places where possible.

1)The probability that wave will crash onto the beach exactly 1.5 seconds after the person arrives is P(x = 1.5) = The probability that it will take longer than 4.06 seconds for the wave to crash onto the beach after the person arrives is P(x > 4.06) =

2)Suppose that the person has already been standing at the shoreline for 0.7 seconds without a wave crashing in. Find the probability that it will take between 2.9 and 5.4 seconds for the wave to crash onto the shoreline.  ________

3)62% of the time a person will wait at least how long before the wave crashes in? ___________seconds.

4)Find the minimum for the lower quartile. _____________ seconds.

Answer 1

X ~ U (0 , 5.8)

1) P(X = 1.5) = 0

P(X > 4.06) = (5.8 - 4.06) / (5.8 - 0) = 0.3

2) P(2.9 < X < 5.4 | X > 0.7) = P(2.9 < X < 5.4) / P(X > 0.7)

                                             = [ (5.4 - 2.9) / (5.8 - 0) ] / [(5.8 - 0.7) / (5.8 - 0)]

                                             = 0.4902

3) P(X > x) = 0.62

or, (5.8 - x) / (5.8 - 0) = 0.62

or, x = 2.204 seconds

4) P(X < x) = 0.25

or, (x - 0) / (5.8 - 0) = 0.25

or, x = 1.45 seconds

lower quartile = 1.45 seconds