Question

Today, the waves are crashing onto the beach every 4.4 seconds. The times from when a person arrives at the shoreline until a crashing wave is observed follows a Uniform distribution from 0 to 4.4 seconds. Round to 4 decimal places where possible. e. The probability that it will take longer than 1.58 seconds for the wave to crash onto the beach after the person arrives is P(x > 1.58) = f. Suppose that the person has already been standing at the shoreline for 0.9 seconds without a wave crashing in. Find the probability that it will take between 1.5 and 3 seconds for the wave to crash onto the shoreline. g. The longest 42% percent of the time a person will wait before the wave crashes is at least how long? seconds. h. nd the minimum for the upper quartile. seconds. ??

Answer 1

Let X be the time until a crashing wave is observed. T ~ Unif(0, 4.4)

e.

P(X > 1.58) = (4.4 - 1.58) / (4.4 - 0) = 0.6409091

f.

P(1.5 < X < 3 | X > 0.9) = P(1.5 < X < 3 and X > 0.9) / P(X > 0.9)

= P(1.5 < X < 3) / P(X > 0.9)

= [(3 - 1.5)/(4.4 - 0)] / [(4.4 - 0.9) / (4.4 - 0)]

= 1.5 / 3.5

= 0.4285714

g.

P(X > x) = 0.42

(4.4 - x)/(4.4 - 0) = 0.42

4.4 - x = 0.42 * 4.4

x = 4.4 - 1.848 =  2.552 seconds

h.

For upper quartile, Q3

P(X < x) = 0.75

(x - 0) / (4.4 - 0) = 0.75

x = 0.75 * 4.4 =  3.3 seconds