Q:

A dolphin show is scheduled to start at 9:00 A.M., 9:30 A.M., and 10:00 A.M. Once the show starts, the gate will be closed. A visitor will arrive at the gate at a time uniformly distributed between 8:30 A.M. and 10:00 A.M. Determine a) The cumulative distribution function of the time (in minutes) between 8:30 A.M. and arrival.

Accepted Solution

A:
Answer:The cumulative distribution function of the time (in minutes) between 8:30 A.M. and arrival is [tex]\frac{x}{90}[/tex] such that 0 < x < 90.Step-by-step explanation:Consider the provided information.A dolphin show is scheduled to start at 9:00 AM, 9:30 A.M and 10:00 A.M.Once the show starts, the gate will be closed. The arrival time of the visitor at the gate is uniformly distributed between 8:30 A.M and 10:00 A.M.The time in minutes is between arrival and 8:30A.M.Uniform distribution is defined as, [tex]f(x) = \frac{1}{b-a}[/tex] where a<x<bHere a=0 and b=90Thus, the probability density function is:[tex]\left\{\begin{matrix}\frac{1}{90}& 0<x<90\\0& otherwise \end{matrix}\right.[/tex]The cumulative distribution function of the time between arrival and 8.30 A.M is,[tex]F(X)=P(X\leq x)[/tex][tex]\int\limits^x_0 {f(u)} \, du \\\int\limits^x_0 {\frac{1}{90}} \, du \\\frac{1}{90}(x-0)\\\frac{x}{90}[/tex]Hence, the cumulative distribution function of the time (in minutes) between 8:30 A.M. and arrival is [tex]\frac{x}{90}[/tex] such that 0 < x < 90.