A company specializes in installing and servicing central...

A company specializes in installing and servicing central-heating furnaces. In the prewinter period, service calls may result in an order for a new furnace. The following table shows estimated probabilities for the number of new furnace orders generated in this way in the last two weeks of September: Number of orders 0 1 2 3 4 5 Probability 0.1 0.14 0.26 0.28 0.15 0.07 Part A - Is the table above a probability distribution? If so, plot it and describe it. Part B - Calculate and graph the cumulative probability distribution Part C - Find the probability that at least 3 orders will be generated in this period

Transcript

00:01 For this problem, i'll be using excel, but i'm not going to be using any sort of advanced functionality within excel, i'll just be using it as a fancy calculator, essentially.

00:09 So i'll first just copy down the data, number of orders and the corresponding probabilities.

00:14 So one of the first things that we are asked, or not just one of the, the first thing that we are asked is whether or not the table shown is a probability distribution.

00:24 So for part a, the first requirement for a probability distribution is that the, or is that, one second here, all of the probability values must be between zero and one, we can trivially see that that is true.

00:40 And then we must have that the sum of all of the probabilities is equal to exactly one.

00:47 So using excel here, i'll just put in equals sum and select all of those values.

00:54 And we can see indeed, the probabilities add up to one.

00:57 So we have then that yes, this is a probability distribution.

01:03 Now this is part of why i'm using excel, we're asked to plot and describe the distribution.

01:08 Of course, you can do this by hand, you know, with a piece of graph paper or something like that.

01:12 But i am not great at any sort of artistic or visual representation.

01:18 So i'm using excel.

01:19 So what we'll do is select all of my values and go to, let's see here, we'll actually go to more column charts.

01:27 And we can see that this one, this clustered column is the one that we want.

01:31 So we'll select that.

01:32 And of course, here, it is pretty appropriate actually to have the gaps between each one of the bars, because we do have the fact that well, it appears that we are dealing with a situation of discrete data, you can't have between zero and one or one and two.

01:49 So it does make a relative amount of sense to have the gaps in between, but that is mostly going to be a matter of taste, essentially.

01:57 But we can see that looking at the distribution, we have that this is actually what we could call a roughly bell -shaped symmetric distribution.

02:07 It's not perfectly bell -shaped and symmetric.

02:11 It's certainly not the sort of thing that we would necessarily call a normal distribution, because of the fact, first of all, it's a completely discrete distribution.

02:19 But we can see that it does have that the results further away from the mean are less common, and the results closer to the mean are more common...

Question

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