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Issues that arise at work are often a reflection of problems occurring in the larger society and the tensions that we're seeing out there today. So today, unfortunately all around the world, we're seeing conflicts, sometimes verbal attacks, sometimes physical attacks. Maybe it's refugees from Syria or other war-torn countries trying to find a place to live in Europe. Maybe it's a tax on immigrants. Maybe in the United States, now it seems to be attacks on religious groups, Muslims. Sometimes now we're seeing racial conflicts rise again. So we thought it would be useful in this course to have a conversation about how these conflicts then affect the workplace, and what we've learned over the years about how to address them effectively. And we're fortunate to have a world-class expert here to draw on. I'm talking today with Professor Mary Rowe, a friend and colleague here in the Sloan School. Mary has been the ombudsperson for MIT for 40 years...

We will now work with a geometric random variable and put to use our understanding of conditional PMFs and conditional expectations. Remember that a geometric random variable corresponds to the number of independent coin tosses until the first head occurs. And here p is a parameter that describes the coin. It is the probability of heads at each coin toss. We have already seen the formula for the geometric PMF and the corresponding plot. We will now add one very important property which is usually called Memorylessness. Ultimately, this property has to do with the fact that independent coin tosses do not have any memory. Past coin tosses do not affect future coin tosses. So consider a coin-tossing experiment with independent tosses and let X be the number of tosses until the first heads. And X is a geometric with parameter p. Suppose that you show up a little after the experiment has started. And you're told that there was so far just one coin toss. And that this coin ...

PROFESSOR: Let's do a work check. So main check. If integral psi star x t0, psi x t0 dx is equal to 1 at t equal to t0, as we say there, then it must hold for later times, t greater than t0. This is what we want to check, or verify, or prove. Now, to do it, we're going to take our time. So it's not going to happen in five minutes, not 10 minutes, maybe not even half an hour. Not because it's so difficult. It's because there's so many things that one can say in between that teach you a lot about quantum mechanics. So we're going to take our time here. So we're going to first rewrite it with better notation. So we'll define rho of x and t, which is going to be called the probability density. And it's nothing else than what you would expect, psi star of x and t, psi of x and t. It's a probability density. You know that has the right interpretation, it's psi squared. And that's the kind of thing that integrated over space gi...

6.004 students work around the dryer bottleneck by finding a laundromat that has two dryers for every washer. Looking at the timeline you can see the plan, which is divided into 30-minute steps. The washer is in use every step, producing a newly-washed load every 30 minutes. Dryer usage is interleaved, where Dryer #1 is used to dry the odd-numbered loads and Dryer #2 is used to dry the even-numbered loads. Once started, a dryer runs for a duration of two steps, a total of 60 minutes. Since the dryers run on a staggered schedule, the system as a whole produces a load of clean, dry laundry every 30 minutes. The steady-state throughput is 1 load of laundry every 30 minutes and the latency for a particular load of laundry is 90 minutes. And now here's the take-home message from this example. Consider the operation of the two-dryer system. Even though the component dryers themselves aren't pipelined, the two-dryer interleaving system is acting like a 2-stage pipeline w...

Now let's work out an example that shows how to use the pairwise independent sampling theorem to actually do some sampling and estimation. So let's remember that our basic theorem says that if we have n independent random variables-- pairwise independent, with the same mean and variance-- and we look at their average, the probability that their average differs from the mean by more than a given tolerance delta is less than or equal to this formula here. Which is the standard deviation over delta squared times 1 over n. Now we're just going to be plugging into this formula, but I want to state it here for the record to remember that this is the pairwise independent sampling theorem that we're given. Which is, what we've seen in general, allows us to calculate the degree of confidence we can have, the probability that we have given n or the n that we need given how confident we want to be. So let's go ahead and do the example. And I want to think abo...