AIHL 4.17 Poisson distribution | Free Mathematics Applications & Interpretation (AI) Video | RevisionDojo
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AIHL 4.17 Poisson distribution Learn AIHL 4.17 Poisson distribution in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 4.17—Poisson distribution.
About this video Learn AIHL 4.17 Poisson distribution in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 4.17—Poisson distribution.
The video provides an overview of the Poisson distribution , which models the probability of a given number of events occurring in a fixed interval of time or space. Key examples include the number of calls received in a call center or the number of patients arriving at a hospital per hour.
Important concepts covered include:
The mean (λ \lambda λ variance of the Poisson distribution are equal.
The distribution is characterized by its independence of events, meaning the occurrence of one event does not affect the probability of another.
Various probability calculations are demonstrated, including finding the probability of specific outcomes and using calculators for Poisson distribution functions.
Additionally, real-life applications of the Poisson distribution, such as its mention in the Bitcoin white paper, highlight its relevance in statistical modeling.
Video transcript 00:00 Hi guys, so in this
00:02 lesson we're going to look
00:03 at the Poisson distribution. Firstly,
00:06 what is it? So the
00:07
Poisson distribution expresses the probability
00:09 of a given number of
00:10 events occurring in a fixed
00:12 interval of time or space,
00:13 right? That sounds complicated, but
00:15 it's not. So some examples.
00:18 The number of calls received
00:20 every hour in a call
00:21 center. The number of buses
00:23 that come every 10 minutes.
00:24 So note guys, you've always
00:25 given a fixed interval of
00:28 the fixed interval times every
00:29 hour here, every 10 minutes.
00:31 Here, the number of potholes
00:33 per kilometer. So it's every
00:34 kilometer in a row. So
00:36 it could be distance. Well,
00:39 it's time or space. It
00:40 could be distance. Could be
00:40 in this case, it's per
00:44 100 millimeter volume. So number
00:46 of white blood cells per
00:48 100 millimeter of blood, something
00:50 like that. So let's kind
00:51 of think, well, this example
00:53 is the number of patients
00:54 that arrive at a hospital
00:58 and it says a Poisson
01:01 distribution with mean 12. So
01:03 the mean 12 means on
01:04 average they get 12 patients
01:08 that occur or that that
01:10 come every hour. Now it
01:11 says here each each event
01:13 is independent of all other
01:14 events. So that's a very
01:15 important assumption. So as soon
01:20 as one patient arrives that
01:21 doesn't change the probability that
01:23 another patient might arrive
01:24 like straight away and then
01:25 another one. And just because
01:27 you've won busy hour doesn't
01:29 mean the next hour is
01:30 hour, etc. So it's all
01:32 independent. That's important. It's it's
01:34 on average 12 patients are
01:36 coming per hour. Now, another
01:39 assumption of this example is
01:40 making is that there's no
01:41 busy hours like there's no
01:43 there's no like I don't
01:46 know evening time 6 p
01:49 .m. It's a busy time
01:50 because I don't know that's
01:52 playing football or something that's
01:53 not taken into account. Okay,
01:57 in the formula booklet we
01:58 have this, it just means
02:01 Poisson distribution M. So there's
02:03 only one parameter for the
02:04 Poisson distribution, only one thing
02:06 you have to worry about
02:08 the mean and the variance,
02:10 they're the same thing, it's
02:11 one of the characters, it's
02:12 the Poisson distribution, the mean
02:14 and the variance are the
02:14 same. Don't worry too much
02:16 about that. You don't need
02:20 the Poisson density function or
02:24 the formula for our calculate
02:27 with our calculator which is
02:29 great. So in this case
02:34 this example. Okay, let's go
02:36 down to this up now.
02:37 Given I've tried got five
02:39 different parts and I've tried
02:40 to kind of cover all
02:41 the different or the main
02:43 different questions that you could
02:44 be asked and obviously IB
02:47 examiners find a way
02:48 is to find different unique
02:51 examples. So make sure you
02:52 practice lots of fast paper
02:53 questions. Okay, so patients number
02:56 of patients arrive at an
02:57 ER per hour. So that's
02:59 important per hour can be
03:01 modeled by a Poisson distribution
03:03 with mean 12. So I'm
03:04 actually going to write here.
03:06 X is Poisson distribution with
03:09 mean 12. Okay, part A.
03:14 Find the probability that eight
03:16 patients arrive between 5 and
03:22 find the probability that x
03:24 equals 8. So the, an
03:27 average 12, I wonder if
03:30 I don't probably probably that
03:31 8 come in this given
03:32 hour between 5 and 6.
03:34 So calculator. Let's go back
03:40 here. I'm going to do
03:42 menu. I'm going to do
03:44 probability or a statistics point
03:47 looking for distribution. So it's
03:49 here in distributions, it's in
03:50 both of them. And now
03:52 Poisson. So you'll find Poisson
03:54 guys. Where is Poisson? The
03:58 Poisson distribution. It is definitely
04:00 right. It's down here at
04:01 the end. Very good. So
04:03 you have Poisson PDF and
04:04 you have a Poisson CDF.
04:05 Yes, we're going to use
04:06 both. It's a bit like
04:08 binomial distribution if you remember.
04:10 PDF is very exactly something
04:16 than 15 for example B.
04:18 Click pause on PDF for
04:19 this example. Lambda, that's my
04:22 M. Usually we use Lambda
04:24 but the formula we click
04:25 choose this M. So my
04:27 Lambda is 12, that's the
04:28 average. And the X value
04:29 is 8, press enter, very
04:32 easy. The probability that this
04:34 happens is 0 .06552 .06552
04:43 it. So there's a low
04:45 probability that you're going to
04:46 get eight patients. But remember,
04:49 that's because there's probabilities for
04:51 everything. So you could get
04:53 seven patients, five patients might
04:55 get zero patients. So that
05:00 seems like a reasonable answer.
05:01 Part B. Find the probability
05:04 that more than 15 patients
05:05 arrive between 5 and 6
05:06 p .m. So the probability
05:13 15. Okay. Same thing, many
05:17 probability distributions, uh, Poisson distribution.
05:21 This time though it's CDF
05:23 because it's more than now.
05:25 Guys, be careful. Lambda is
05:27 12. What's my lower bound?
05:31 Well, if he says more
05:32 than 15, we're not counting
05:34 15. So, well, this isn't
05:36 for a normal distribution, because
05:38 it's continuous. This is important
05:40 for a post -undistration, because
05:42 post -undistration is a discrete
05:43 distribution. It's not continuous. So
05:46 at my lower bound is
05:48 16. And my upper bound
05:50 is 999999, because you could
05:53 guess an infinite number of
05:55 people. Guys, this is definitely
05:58 definitely a big enough number
06:01 but whatever. Press OK, 0
06:04 584, 0 .155584. So there's
06:14 a chance, 15 % chance
06:15 or 16 % chance, that
06:18 you're going to get more
06:19 than 15 patients, which again
06:21 is kind of what you'd
06:24 expect, right? See, find the
06:28 probability that at least four
06:30 patients arrive between six and
06:33 I'll be careful guys, it's
06:34 changed, it's now half an
06:36 hour. So what we can't
06:39 do, you can't just find
06:41 the probability that it's for
06:47 half it, it doesn't work
06:48 like that. You have to
06:49 change the distribution. So you
06:51 have to actually write Y,
06:55 Poisson distribution. But with mean
07:01 this half the mean. It
07:02 is scalable. So you can
07:04 change this for whatever the
07:06 time frame. You can change
07:08 this, but just, well, keeping
07:12 they wanted to figure out
07:14 some problem in, I don't
07:16 know, 10 hours, it's a
07:19 Poisson distribution with mean 120.
07:22 hour, I'm gonna skip it.
07:25 Sorry, I'm gonna half it,
07:29 Okay, and now I'm trying
07:30 to find the probability that
07:32 say? At least four patients
07:34 arrive. So at least four
07:36 means greater or equal to
07:38 four. So the same thing
07:41 guys, probability, many, probability distributions.
07:47 Pause on CDF, because it's
07:49 greater or equal to. Now
07:52 lower bound is four, because
07:55 my upper bound, again, is
07:57 and 999, press enter, and
08:09 look, we're probably going to
08:10 get at least four patients.
08:13 It's unlikely you're going to
08:14 get less than four patients
08:18 Okay, Part D, right? This
08:20 is where it gets tricky.
08:21 Given that, so once you
08:23 see that guy's given that,
08:25 conditional probability. So given that
08:27 less than 12 patients arrive
08:28 between 4 and 5, find
08:30 the probability that more than
08:31 7 patients arrive between 4
08:33 and 5. That sounds complicated
08:34 because our conditional probability always
08:37 go back to this formula.
08:38 The probability of A given
08:41 B is the probability of
08:47 probability of B. This is
08:48 given in the formula of
08:49 a bucket. It's both of
08:50 them happening, divided by the
08:53 Find the probability that more
08:55 than seven patients arrive. So
08:57 I want the probability that
08:59 x is more than seven.
09:02 Because more than five the
09:03 probability that more than seven
09:04 patients arrive. And we're dealing
09:05 with one hour because it's
09:07 between four and five. Given
09:10 that, given that less than
09:14 12 patients arrive. So given
09:21 about it guys, it's the
09:22 probably the next is bigger
09:23 than 7, given the next
09:24 is less than 12. And
09:26 this equals to the probability
09:28 of both of them happening.
09:32 than 7 and less than
09:35 the probability of both of
09:36 those things happening? Well, it's
09:38 exactly that. It's the probability
09:43 Don't go multiplying the probabilities
09:45 or anything. It's just this
09:48 is between 7 and 12.
09:49 We can calculate that on
09:51 our calculator. And then divide
09:54 it by the probability of
09:55 b, so I'm dividing by
09:56 the probability that x is
09:58 less than 12. Divide by
10:02 the given. So, let's do
10:06 this. Between seven and 12.
10:11 So, let's do that one
10:11 first. Menu, probability, distributions, normal
10:17 Lambda is 12 guys because
10:20 we're dealing with one hour
10:23 the lower bound is 8
10:26 lower bound is 8 and
10:28 than 12 so the upper
10:31 very careful that guys. So
10:42 denominator is less than 12
10:45 many probability distributions. CDF lambda
10:51 is 12. The lower bound
10:54 is zero because that's the
10:57 smallest it can be. And
10:58 the upper bound is 11
11:01 because it has to be
11:01 less than 12. Press okay.
11:05 That's 0 .461597. 0 .46159
11:13 9 7. Nearly there. And
11:19 this equals guys, let's just
11:21 do this properly. It's 0
11:24 .37293 divided by 0 .461597.
11:35 Press enter 0 .806099 0
11:41 Oh, nine, nine. Okay. Nearly
11:48 done. Part E, let's do
11:49 it over here. It says,
11:52 if more than 18 patients
11:53 arrive in a given hour,
11:55 the ER is considered very
11:58 busy for that hour. I
11:59 find, let's just the way
12:00 they've defined it as a
12:01 very, they're saying it's very
12:02 busy for that hour. Find
12:04 the expected number of very
12:05 busy hours in a given
12:06 day. So more than 18
12:08 patients, what's the probability
12:12 18. We've done this already.
12:14 We've done this type many
12:16 probability distributions, CDF 12 more
12:21 than 18. So the lower
12:22 bound is 19 upper bound,
12:25 999, 999, 0 .037416, 0
12:33 .037416. So they're not going
12:37 get many, not going to
12:41 get many very busy hours.
12:44 But let's find the expected
12:45 number. So how many hours
12:50 many of those hours are
12:50 going to be very busy?
12:55 So I'm just going to
13:01 And that gives me, and
13:05 that's just your time.
13:06 24, 0 .897996, 0 .8979689979796996.
13:19 So that is the expected
13:23 number of very busy hours.
13:25 Yes, it's less than one.
13:26 Should I round to one?
13:28 No, not for expected. Not
13:31 for expected value because it's
13:32 like the mean. Okay.
13:34 So that's it guys, that's
13:35 the post -industrial distribution. I
13:39 think this example kind of
13:41 covers most of the types
13:43 of post -industrial question you
13:45 can be asked. There are
13:46 tests and statistical tests that
13:49 later videos that involve the
13:51 post -industrial distribution. And certainly
13:52 you need to know how
13:53 to do them. There are
13:55 many real life, very interesting
13:57 real life applications of the
13:59 post -industrial region and I
14:00 want to just show you
14:02 Hopefully it's here. Yeah, the
14:04 Bitcoin white paper. This was
14:07 actually if you click on
14:08 this available obviously forever into
14:11 read. This is actually how
14:12 Satoshi Nakamoto introduced Bitcoin to
14:16 everyone and this is actually
14:18 not that a complicated complicated,
14:21 but it's not like that
14:23 complicated to read, but right
14:24 down at the bottom. I
14:25 think it's only nine pages
14:28 right down at the bottom
14:30 Here, you'll see that he
14:34 uses the Poisson distribution to
14:38 show how basically how it's
14:42 incorruptible. I'm not going to
14:44 go into detail about this
14:46 obviously right now, but I
14:47 thought it was a nice,
14:49 interesting, real life application of
14:50 the Poisson distribution. Okay, that's
14:54 it guys. I'll see you