### Probability Calculation of Play Completion TimeConsider a play that is being performed regularly at a local theatre. The duration of the play and its intermission can both be described using normal distributions.#### Given Data:1. **Play Time:** - Mean (\(\mu_{\text{play}}\)): 130 minutes - Standard Deviation (\(\sigma_{\text{play}}\)): 3 minutes 2. **Intermission Time:** - Mean (\(\mu_{\text{intermission}}\)): 15 minutes - Standard Deviation (\(\sigma_{\text{intermission}}\)): 5 minutesWe aim to find the probability that the total performance (play plus intermission) is completed within 140 minutes.### Aggregated Distribution of Performance TimeDefine:- \(X\): Random variable representing the play time- \(Y\): Random variable representing the intermission timeThese variables are independently normally distributed:- \(X \sim N(130, 3^2)\)- \(Y \sim N(15, 5^2)\)The total performance time \(Z\) is the sum of \(X\) and \(Y\):\[ Z = X + Y \]The mean (\(\mu_Z\)) and variance (\( \sigma_Z^2 \)) of \(Z\) are given by:\[ \mu_Z = \mu_{\text{play}} + \mu_{\text{intermission}} = 130 + 15 = 145 \]\[ \sigma_Z^2 = \sigma_{\text{play}}^2 + \sigma_{\text{intermission}}^2 = 3^2 + 5^2 = 9 + 25 = 34 \]\[ \sigma_Z = \sqrt{34} \]Thus, \(Z\) follows a normal distribution:\[ Z \sim N(145, \sqrt{34}) \]### Probability CalculationWe need to calculate:\[ P(Z < 140) \]### Standardizing the Normal VariableConvert \(Z\) to the standard normal variable \(Z'\):\[ Z' = \frac{Z - \mu_Z}{\sigma_Z} = \frac{140 - 145}{\sqrt{34}} = \frac{140 - 145}{5.831} = -0.857 \]### Using the Standard Normal Distribution TableDetermine the probability corresponding to

We have given that, Play time be modelled are normally distributed with mean (Mu) = 130 minutes…

3 A play is enjoying a long run at a theatre. It is found that the play time may be modelled as a normal variable with mean 130 minutes and standard deviation 3 minutes, and that the length of the intermission in the middle of the performance may be modelled by a normal variable with mean 15 minutes and standard deviaton 5 minutes. Find the probability that the performance is completed in less than 140 minutes.

3 A play is enjoying a long run at a theatre. It is found that the play time may be modelled as a normal variable with mean 130 minutes and standard deviation 3 minutes, and that the length of the intermission in the middle of the performance may be modelled by a normal variable with mean 15 minutes and standard deviaton 5 minutes. Find the probability that the performance is completed in less than 140 minutes.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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