# Class Notes for 18.44 at Massachusetts Institute of Technology (MIT)

• 28 Results
MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 13: Mit Opencourseware

OC4822923 Page
11 Jun 2015
28
The doob"s optional stopping time theorem is contained in many basic texts on probability and martingales. (see, for example, theorem 10. 10 of. The es
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 13: Prati, Ratia, Royal Institute Of Technology

OC9068084 Page
22 May 2016
8
Break choosing one of the items to be counted into a sequence of stages so that one always has the same number of choices to make at each stage. Then t
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 11: Random Variable

OC9068081 Page
22 May 2016
4
Lecture 11: binomial random variables and repeated trials. Toss fair coin n times. (tosses are independent. ) Can use binomial theorem to show probabil
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 10: Dried Lime, Elche, Random Variable

OC9068081 Page
22 May 2016
7
The variable of x, denoted var(x), is de ned by var(x) = e[(x - ) ] Taking g(x) = (x - ) , and recalling that e[g(x)] = (x - ) p(x), we nd that var[x]
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture 1: 18.600 Feb 3, 2016 (Lecture 1)

OC9068081 Page
22 May 2016
7
Suppose that betting markets place the probability that your favorite presidential candidates will be elected at 58 percent. Price of a contract that p
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 22: Internet Message Access Protocol, Independent And Identically Distributed Random Variables, Random Variable

OC9068081 Page
22 May 2016
8
Say we have independent random variables x and y and we know their density functions f and f. Now let"s try to nd f (a) = p{x + y a} This is the integr
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 19: Random Variable, Press Kit, Ert1

OC9068082 Page
22 May 2016
6
Say x is an exponential random variable of parameter when its probability distribution function is f(x) = For a > 0 we have f (a) = f(x)dx = e dx = - e
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 18: Plat

OC9068082 Page
22 May 2016
9
Let"s plot this for a few values of n. If we replace fair coin with p coin, what"s probability to see k heads. Let"s plot this for p = 2/3 and some val
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 30: Light-Year, Winny

OC9068082 Page
22 May 2016
6
Consider a sequence of random variables x , x , x , . each taking values in the same state space, which for now we take to be a nite set that we label
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 16: Random Variable, Public Radio Exchange

OC9068082 Page
22 May 2016
2
Say x is a continuous random variable if there exists a probability density function f = f on such that. We may assume f(x)dx = f(x)dx = 1 and f is non
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 14: Random Variable, Stochastic Process, Philippine Standard Time

OC9068082 Page
22 May 2016
9
Properties from last time for integer k 0. A poisson random variable x with parameter satis es p{x=k} = The probabilities are approximately those of a
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MIT18.44Scott SheffieldSpring

## 18.44 Lecture Notes - Lecture 9: Weighted Arithmetic Mean, Narn, Riak

OC9068082 Page
22 May 2016
4
Recall: a random variable x is a function from the state space to the real numbers. Can interpret x as a quantity whose value depends on the outcome of
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