WEBVTT mathematics/probability/murray
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Hi and welcome back to the probability lectures here on www.educator.com, my name is Will Murray.
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Today's lecture is on Markov’s inequality.
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Markov’s inequality is one of two inequalities that you can use to estimate probabilities quickly,
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the other one was Tchebysheff's inequality.
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The next lecture will be on Tchebysheff's inequality.
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If you are looking for that one, just skip ahead on the next video.
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If you are looking for Markov’s inequality, here we go.
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Markov’s inequality is a quick way of estimating probabilities based only on the mean of a random variable.
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You all you have to know is the mean of a random variable or the expected value.
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Remember that mean and expected value mean exactly the same thing.
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One important condition that you need to use Markov’s inequality is that, your random variable has only positive values.
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You have to be estimating things that can only be counted in positive numbers, like a number of customer at a business,
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or the number of miles a car is driven, or the amount of money you have assuming you are not allowed to go into debt,
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things like that, that can be only measured using positive values.
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If it is a random variable that takes on negative values then Markov’s inequality is not necessarily true.
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The way it works is you have some constant number, that is this value A here.
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What we are going to do is estimate the probability that the variable will be bigger than that value A.
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Markov’s inequality gives you an answer for that.
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It says that the probability is less than the expected value of Y or that is the same as the mean divided by A.
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Maybe another form you might have seen this inequality is μ/A
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because remember that people use μ as the expected value as a shorthand for the expected value.
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There is one thing I want to emphasize about Markov’s inequality which is that it is really a one sided estimation.
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This is a one sided bound, it gives you an upper bound.
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It does not tell you that the probability is equal to that, it just gives you an upper bound, one sided bound on the probability.
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Whenever you answer a question using Markov's inequality,
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your answer will always be something like the probability is less than something or the probability is greater than something.
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You can never say that the probability is exactly equal to something based on Markov’s inequality.
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It just gives you a one sided upper bound.
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You can reverse this, we said we are calculating the probability that Y is greater than or equal to A to the basic form of Markov’s inequality.
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You can also switch that around and ask what the probability of Y being less than A?
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That is exactly the opposite of Y being greater than or equal to A, we get the complement of that.
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The probability is now greater than or equal to 1 - the expected value of Y/A.
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Remember, we can also write that as the expected value of Y as μ.
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You can say it is greater than or equal to 1 - μ/A.
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Again, this is a one sided thing.
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It can tell you that the probability that Y is greater than A is less than something.
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It can tell you that the probability that Y is less than A is greater than something, but you can never reverse those.
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You have to be very careful about how you use Markov’s inequality which way into inequality go,
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more practices as we go through some of the examples.
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You also have to be careful never to say this probability is equal to something
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because Markov’s inequality will never tell you that.
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It will always just say this probability is less than something or this probability is greater than something.
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Let us check this out with some examples.
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First example, have we done a survey on a particular college campus and apparently,
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the students on this campus are all going to have some cash in their wallets.
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It turns out that the average amount of cash that these students are carrying is $20.00.
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The question is, if we meet a student at random and ask here how much cash she is caring,
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what is the chance that it should be carrying more than $100?
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Let us estimate the chance that she is carrying less than $80.00.
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Let me first emphasize that this is a situation in which Markov’s inequality does apply
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because the amount of cash students are carrying is always going to be a positive amount.
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The smallest you can be caring is 0.
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You could be carrying a definite amount of cash.
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You could be carrying thousands of dollars in cash but you cannot be carrying a negative amount of cash.
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This is always the amount of cash that you are carrying is always positive.
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It is a situation in which we can apply Markov’s inequality.
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Let me go ahead and write down Markov’s inequality and we will see how to apply it to the situation.
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Markov’s inequality, remember, we said the probability that Y is greater than
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or equal to a certain value A is less than or equal to the expected value of A, the mean of Y divided by A.
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In this case, the first question here we want to estimate the chance that a student is carrying more than $100.
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We want to find the probability that a student will have more than 100 and I’m going to fill in the expected value of Y,
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that is the average value of Y which we have been given as $20.00.
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It is 20 for the expected value and then the value of A that we are using is 100, 20/100 simplifies down to 1/5.
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The answer to our first question there is the probability is less than or equal to 1/5.
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Notice that I'm being very careful here not to say that it is equal to 1/5.
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It might be considerably less than 1/5.
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I do not know if it is less than or less or equal to.
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To be safe, I’m just going to say the probability is less than or equal 1/5.
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That is really all we can tell using Markov’s inequality.
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We also want to estimate the chance that a student is carrying less than $80.00.
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That is the other direction of Markov’s inequality, the probability that Y is less than the value A.
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Reversing Markov’s inequality, that probability is greater than or equal to 1 – E of Y, the expected value /A.
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That is the equation that we learned back there on the first side.
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In this case, our A is 80.
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The probability that Y is less than 80 is greater than or equal to 1 -, the expected value is still 20, that was given in the problem.
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20/80 is ¼, this is equal to 1 – 1/4 which is ¾.
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The probability is greater than or equal to ¾.
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What we can say there, if we meet a random student is that the probability that she will have less than $80.00 is at least ¾.
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At least 75% chance she has less than $80.00 in cash on her.
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That answers both of our questions here.
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Note that I could not give you exact probabilities.
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In either case, I have to give you just inequality because that is all Markov’s inequality gave you.
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Let me remind you how we did that.
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I start off with the basic formula of Markov’s inequality.
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This is just the same equation we got on the first slide here.
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Since, we are asked about carrying more than $100, I filled in A =100 here.
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A equals 100 here and the expected value that is the average of $20.00.
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I plugged that in right here and then I just simplify that down 20/100 is 1/5.
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It is important to get the inequality the right way.
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What we can say here is that it is unlikely that a student will have more than $100 and
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how unlikely it is less than 1/5 or less than 20% chance.
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On the other side, we are asked about the chance that she is carrying on less than $80.00.
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I’m using the less than form of Markov’s inequality, that was the second version that I gave you.
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We were told that A equals 80, we plug in the expected value is 20.
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Simplify that down to ¾ and what I can tell you is that if I meet a student,
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there is at least a 75% chance that she will have less than $80.00.
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Let us keep that going with the next example here.
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Here we have a rental car agency, they are doing some statistical analysis of their cars.
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They require that customer return cars after a weeks rental, they put an average of 210 miles on the cars.
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We just had a new customer check out a car for a week and we want to estimate the probability
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that the customer will put more than 350 miles on the car.
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This is a classic Markov’s inequality problem, let me write down Markov’s inequality to get us started.
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The probability that Y is greater than or equal to A is less than or equal to the expected value or the mean value of Y/A.
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In this case, we want to estimate the probability that he will put more than 350 miles.
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The 350 is the A there, 350 is less or equal to.
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The expected value is the average number of miles that these customers are putting on the cars, that is 210/350.
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If I divide top and bottom by 70 there, we also have a factor of 70, that would simplify down to 3/5.
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That is as simple as it is going to get.
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The probability is less than or equal to, 3/5 is 60%.
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I will write that as a percentage.
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What that tells me, what we can tell our associates for this rental car company is that this particular customer,
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there is less than 60% chance that this customer is going to put 350 miles on the car.
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That is the best we can say, we can never give an exact answer with Markov’s inequality.
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We can just put a bound on it, above and below.
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Here, we put an upper bound of 60% chance that the customer is going to put that many miles on the car.
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To show you how I got that, start out with the basic version of the Markov’s inequality.
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I figured out that the A I was looking for was 350, that came from the stem of the problem here.
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I plugged that in, 350 in both places.
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210 is the average number of miles the customers put on the car, that is the expected value of the random variable.
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That simplifies down to 3/5 and the important thing here is that you give your answer as an inequality.
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You do not want to just say 60%.
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When I taught probability, a lot of times my students will just try to give me a number as an answer.
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This 60% as the answer and it does not tell me what I want to know because you are saying that is equal to 60%.
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And we do not know that, all we know is that it is less than or equal to 60%.
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That is all Markov’s inequality tells us.
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In example 3 here, we have done some tracking of the history of earthquakes in California.
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Apparently, there is a major earthquake in California on average, once every 10 years.
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We want to describe the probability that there would be an earthquake in the next 30 years.
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Maybe, we are planning a major investment in California and we are wondering
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how likely it will be that there will be an earthquake in the next 30 years.
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Let me described carefully here what the random variable is,
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because I think it is a little less obvious in this one than in some of the previous ones.
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Y here is going to be the waiting time until the next major earthquake.
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What they have really told us when we say that it occurs on average once every 10 years,
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they told us that the average waiting time for one earthquake to the next is 10.
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E of Y is equal to 10, that is what they have given us.
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We want to find the probability that there will be an earthquake in the next 30 years.
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The probability that our waiting time for the next earthquake is less than 30, that is what we are trying to calculate.
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Within the next 30 years is what we are trying to find.
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We are going to use Markov’s inequality but since we are trying to estimate the probability that Y is less than a cut off,
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we are going to use the version of Markov’s inequality, the reversed version of Markov’s inequality,
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P of Y less than A is greater than or equal to 1 – P of Y/A.
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In this case, our A is 30.
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This is 1 – E of Y which is 10/A is 30.
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I put that that was equal to, I have committed the crime in Markov’s inequality
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that I have been telling you not to do which I said the probability was equal to something.
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We never know that for sure, not for Markov’s inequality.
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We always get a one sided bound so the probability is greater than or equal to 1 – 10/30
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and that simplifies down to 1 -1/3 which is 2/3.
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What we can say here is that the probability that Y is less than 30, remember,
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that is the probability that we will have an earthquake in the next 30 years is greater than or equal to 2/3.
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That is the conclusion we can make from the information we are given and from Markov’s inequality.
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What that saying is that it is pretty likely that there will be an earthquake in California sometime in the next 30 years, or at least a 2/3,
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or 67% chance that there will be an earthquake in California in the next 30 years.
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To show you how I figure that out, the important thing here was setting up the random variable.
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We said Y is going to be the waiting time, how long the wait until we see the next major earthquake.
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Since, they occur once every 10 years on average, that does not mean they occur with clockwork regularity every 10 years.
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It just means they occur on average, once every 10 years.
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The expected value of that variable is 10.
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We want to find the probability that it is less than 30 because if it is less than 30
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that means we will have an earthquake in the next 30 years sometime.
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According to reversed formula for Markov’s inequality, that is a bigger than 1 - the expected value/30.
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Remember, I use equals and that was a mistake, it is really greater than or equal to, that simplifies down to 2/3.
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Our final conclusion here is that the probability is greater than 2/3, greater than or equal to 2/3.
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I think there is at least a 67% chance that we will have an earthquake in the next 30 years here in California.
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For example 4, we have a factory that produces batches of 1000 laptops.
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I guess each day they runoff a batch of 1000 laptops and send them out for distribution.
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They find that on average they will do some testing, on average 2 laptops per batch are defective.
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They have some kind of a serious defect in them.
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We want to estimate the probability that in the next batch, fewer than 5 laptops will be defected.
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Again, this is a Markov’s inequality problem.
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Let me go ahead and set up the generic inequality for Markov.
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That is the probability that Y is greater than or equal to A is less than or equal to the expected value of Y divided by A.
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In this case, we want to reverse that because we want estimate the probability that fewer than 5 laptops will be defective.
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Let me go ahead and do the reverse version of that.
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The probability that y is less than A is greater than or equal to 1 – E of Y/A.
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We are just taking Markov’s inequality and then taking the complement of it.
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That should not be something you really have to memorize, it should be something you can figure out from the original Markov’s inequality.
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In this case our A is 5, we want the probability that fewer than 5 laptops will be defective.
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It is greater than or equal to 1 -, E of Y is the expected value or the mean or the average number of laptops per batch.
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We are given that that is 2, this is 1 - 2/5 here.
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I'm filling in 5 for the value of A because that is what we had on the left hand side.
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1 - 2/5 is 3/5 and we could simplify it, we can convert that into a percentage.
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The probability here is greater than or equal to 3/5 is 60%.
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If you are a company manager, and you got some quality control specifications
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that says you cannot have any more than 5 laptops per batch be defective,
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which you can say is that in the next batch, there is at least a 60% chance that we would not have 5 or more laptops defective.
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That is the best you can say with Markov’s inequality.
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You cannot put a precise value on the probability, you can just give a lower bound and say
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at least 60% of the time that we will have more than 5 laptops be defective.
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Where that came from was, I started with the original version of Markov’s inequality and then
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I realize that I needed to turn this around because the original version has Y being bigger than the cutoff A.
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In this case, I want to estimate the probability that fewer than 5, that is why less than A.
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That is the reverse of Markov’s inequality.
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The probability that Y is less than A is greater than or equal to 1 – E of Y/A.
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Plug in A equals 5 because that is coming from the stem of the problem.
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I plugged in the expected value that is the average number of laptops per batch, where that come from, that comes from here.
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That is where that 2 come from.
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Let me simplify the numbers down to 3/5 which is 60%.
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What I can say from this is that the probability is at least 60% that fewer than 5 laptops are defective in the next batch.
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If you are the factory manager, you can decide whether that is acceptable.
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Are you willing to accept the 60% chance of having fewer than 5 defectives
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or do you need to tighten your quality control procedures based on that probability?
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Let us keep moving onto the next example here.
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In our final example, we got a grocery store that selling an average of 30 cans of tuna per day.
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We want to estimate the probability that it will sell more than 80 cans tomorrow.
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You are the manager of this store and you are worried about whether you are going to run out of your stock of cans of tuna tomorrow
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Should you order some more or can you hold out for a couple more days?
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Again, this is kind of a classic Markov’s inequality problem.
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Something that makes it Markov’s inequality, I did not mention this on some of the previous examples,
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is that we are only calculating values that are going to be positive here.
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The number of cans of tuna that a grocery store is going to sell in any given day, that is going to be positive.
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It could be 0, it could be significantly higher than 0 but it is almost certainly not going to be negative.
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We are not going to be having many cans of tuna return.
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This is a positive quantity, it is okay to use Markov’s inequality.
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Our Y here is the number of cans of tuna sold each day.
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Let us write down our Markov’s inequality.
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The probability that Y is less than or equal to A is greater than or equal to E of Y divided by A.
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That is just our generic formula for Markov’s inequality, we learned that back in the first slide of this lecture.
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In this case, our A is our cutoff value.
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The probability in this case, we are trying to estimate the probability that it will sell more than 80 cans.
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I wrote down my Markov’s inequality, I wrote the inequality so I got them switched here.
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Probably, the Y is greater than or equal to A is less than or equal to E of Y/A, that is the original version of Markov’s inequality.
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The one that I was kind of channeling there was the opposite one,
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the probability that Y is less than A is greater than or equal to 1 - E of Y/A.
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We have to look at our problem and figure out which one of those is going be relevant.
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In this case, we want the probability that it will sell more than 80 cans tomorrow so that is a greater than or equal to.
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The probability that Y is greater than or equal to 80 is less than or equal to
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the expected value not the average number of cans it sells on a normal day, that is 30 cans.
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Let me fill in 80 here for my value of A.
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If I simplify 30/80 that this reduces to 3/8.
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If you convert that into a percentage, that is very easy to convert into a percentage.
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It is halfway between 1/4 and ½.
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It is halfway between 25% and 50% , that is a 37.5%.
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The probability is less than or equal to 37.5%.
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If you are a store manager and you are wondering, maybe, you got 80 cans of tuna in stock.
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You are worry about whether you are going to sell out tomorrow, maybe you are going to need to order some more.
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The probability that you are going to sell all 80 of those cans is at most 37.5%.
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You do not know exactly what the probability is, because Markov’s inequality never tells you
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an exact value but it tells you that it is less than or equal to 37.5%.
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Let me recap how we did that.
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We started with the basic version of Markov’s inequality, P of Y is greater than or equal to A less than E of Y/A.
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I went ahead and wrote down the reverse version, because when I wrote down the basic version,
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I accidentally switched the inequalities.
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I wrote down the reverse version just to make sure that we are keeping everything straight there.
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Since we are selling more than 80 cans that means we want the positive version of Markov’s inequality, the more than version.
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I fill that in with A equals 80, filled in A = 80 in the denominator here.
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The E of Y that is the average number of cans sold, that is the 30 from the problem stem.
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That simplifies down to 3/8 or 37.5%.
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What Markov’s inequality tells us is that the probability is less than or equal to 37.5%.
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That less than or equal to is really an important part of your answer.
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You are giving an upper bound, you are not saying it is equal to 37.5%.
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You are just saying that is the most it could possibly be.
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That wraps up our lecture on Markov’s inequality.
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Next, we are going to be talking about Tchebysheff's inequality which is a little bit stronger than Markov’s inequality.
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You usually get a stronger version with Tchebysheff's than Markov
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but after a little bit work form to Tchebysheff's inequality, you have to know the standard deviation.
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Markov’s inequality, we just had to know the expected value or the mean of the random variable.
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I hope you stick around and we will learn about Tchebysheff's inequality in the next video.
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In the meantime, you have been watching the probability videos here on www.educator.com.
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My name is Will Murray, thank you for watching today, bye.