As beowulf tries to make himself coffee in the common area...

As Beowulf tries to make himself coffee in the common area, he notices that the sole machine is damaged no longer works. If it takes 2 minutes for the machine to brew a cup, following an exponential distribution, and 20 people per hour want coffee, following a Poisson-distribution, calculate the following: A. The percentage of time that the machine is busy. B. The average number of people waiting in line. C. How long it takes on average to get a cup of coffee. D. The probability that no one is waiting in line for coffee

Transcript

00:01 There is given in normal distribution in this question and the mean value is given.

00:05 Let me just write down here the mean which is denoted by moon that is 125 seconds.

00:11 And the standard deviation, which is denoted by sigma, that is 22 seconds.

00:18 Since this is normal distribution, we can define the random variable x, which is normally distributed.

00:23 That is 125 and 22.

00:25 So in part a, we have to just find the percentage that less than 90 seconds.

00:31 So the random variable x which is less than 90 so what we have to do so that means for example if we just just draw a normal curve here the mean value is here which is 125 and the 90 is here so we need to get the area of this yellow shaded region so what we have to do we are going to use the normal cdf function of the calculator when we look at the shaded region that goes to negative in here so i'm going to put negative 1 in 99 and the upper boundary is 90 and the mean value is 125 and the standard division is 22 let's get the value from the calculator first of all we have to just press the button second here and then the distribution there is normal cdf here so the lower boundary negative 1 this is 99 and the upper boundary is 90 and the mean value is 125 in the standard 2.

01:33 Great.

01:34 We got the answer as 0 .05 and 58.

01:37 0 .05 and 58.

01:41 So we have to give the answer in width percentage to two decimal places.

01:46 Let's multiply with 100%.

01:49 We got 5 .58 % of the values are less than 90 seconds here.

01:56 And let's take a look at part b.

01:59 What percentage of weight times are greater than 125.

02:04 So the probability of x is greater than 125.

02:07 Again, we're going to use the normal cdf function here.

02:12 The lower boundary is 125, but in this case, the upper boundary is that goes to positive infinity, so 1e99, and the mean value is 125 and the standard division 22.

02:24 Then we'll look at this shaded region.

02:26 So this shaded region is the half of the total area.

02:29 So the total area is one, and the half of this one is one half, so which is equal to 0 .5.

02:35 So that means this is 50 % of the values are greater than 125 seconds.

02:42 So let's take a look at the next part of the question.

02:47 So it says what percentage of times those are between 90 and 25 seconds? so in part c, we have to find the random variable x, which is less than 125, and greater than 90...

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