A movie theatre scheduled three sessions for a movie at: 5:00pm, 7:00pm and 9:00

Question

A movie theatre scheduled three sessions for a movie at: 5:00pm, 7:00pm and 9:00pm. Once the movie starts, the gate will be closed. A visitor will arrive at the movie theatre at a time uniformly distributed between 4:00pm and 9:00pm. (i.e., t~U[4:00pm, 9:00pm]) Determine the cumulative distribution function of the arrival time in minutes Use the cumulative distribution function to determine the following: The probability that the visitor attends at least one movie session The probability that the visitor waits more than 20min for a movie session The probability that the visitor waits less than 10 min for a movie session The probability that the visitor attends the second show knowing that he missed the first show.

Explanation / Answer

here arrival time is unifomly distribute (a,b) with a =0 minutes and b =5*60=300 minutes

1) CDF of the arrival time =P(X<x) =(X-a)/(b-a) =(x/300)

2) the probabilty that visitor attends at least one movie session =P(arrive before 9 p,m)=(300/300)=1

3) probabilty that visitor waits more then 20 minutes=P(arrive b/w 4:00 to 4:40 +arrive b/w 5:00 to 6:40+arrive b/w 7:00 to 8:40) =240/300=0.8

4) ) probabilty that visitor waits less then 10 min =P(arrive b/w 4:50 to 5:00+arrive b/w 6:50 to 7:00+arrive b/w 8:50 to 9:00)=30/300 =0.1

5) P(5:00 PM <X<7:00PM)/P(X>5:00 PM) =120/240=0.5

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