CS 237: Probability in Computing

Transcription

1 CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 13: Normal Distribution Exponential Distribution

2 Recall that the Normal Distribution is given by an explicit formula, using the mean and variance/standard deviation as parameters: where! = mean/expected value " = standard deviation " 2 = variance 2

3 The normal distribution, as the limit of B(N,0.5), occurs when a very large number of factors add together to create some random phenomenon. Example: Even REALLY IMPORTANT things are normally distributed!

4 Recall that the only way we can analyze probabilities in the continuous case is with the CDF: P(X < a) = F(a) P(X > a) = 1.0 F(a) P(a < X < b) = F(b) F(a) 4

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12 Standard Normal Distribution Since there are a potentially infinite number of Normal Distributions, sometimes we calculate using a normalized version, the Standard Normal Distribution with mean 0 and standard deviation (and variance) 1: Any random variable X which has a normal distribution N(μ,σ 2 ) can be converted into a N(0, 1) standardized random variable X* with distribution N(0,1) by the usual form. In the case of the normal distribution this is labelled as Z: This is usually helpful in HAND calculations, since μ and σ have been factored out... 12

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17 If you were doing these calculations in 1900, or haven t heard of a computer, or if you were taking a test where you didn t have a calculator, here is how you would calculate the probability of a normallydistributed random variable: 17

18 Modern people use the appropriate formulae: 18

19 Or a calculator or a web site: 19

20 Recall: Poisson Process Formally, we have the following definition: suppose we have discrete events occurring through time as just described, and let such that 1) The expected value of N[s..t] is proportional to the length (t s) of the interval; in particular, for any two non-overlapping intervals of the same length, the mean number of occurrences in each is the same; 2) The number of arrivals in two non-overlapping intervals is independent; and 3) The probability of two events occurring at the same time is 0. Then this random process is said to be a Poisson Process. We shall be dealing only with discrete intervals of time for now, and so the important things to remember are that the intervals are independent and the mean number of arrivals in each time unit is the same over its infinite range

21 Recall: Poisson Random Variables Suppose we have a Poisson Process and we fix the unit time interval we consider (say, 1 second or 1 year, etc.), where the mean number of arrivals in a unit interval is!, and then each time we poke the random variable X we return N[0..1], N[1..2], N[2..3], etc. Then we call X a Poisson Random Variable with rate parameter!, denoted where

22 Interarrival Times of a Poisson Process Suppose we have a Poisson Process, and instead of counting the number of arrivals in each unit interval, we look at the interarrival times, i.e., the amount of time between each arrival. Intuitively, this is a natural thing to think about: How long before the next event? Y Let s define the random variable Y = the arrival time of the first event. In fact, because the arrivals are independent, at any time t, probabilistically the Poisson process starts all over again (the events don t remember the past!), so in fact: Y = the interarrival time between any two events Now the question is: What is the distribution of Y?

23 Interarrival Times of a Poisson Process Y What is the distribution of Y? Since and the number of arrivals in an interval is proportional to its length, that is, E( N[0..2] ) = 2 * E( N[0..1] ), etc., then and so the probability that there are n arrivals by time t is and

24 Interarrival Times of a Poisson Process Y What is the distribution of Y? Recall the chain rule: Now, this is the formula of a CDF, that is, and so if we take a derivative, we get the PDF:

25 Exponential Distribution This is called the Exponential Distribution, and along with the Normal, is one of the most important continuous distributions in probability and statistics. Formally, then, if the random variable Y = the interarrival time between events in a Poisson Process we say that Y is distributed according to the Exponential Distribution with rate parameter!, denoted if and where and

26 Exponential Distribution: The Memoryless Property m n+m n The exponential, like the geometric, has the memoryless property, n and the proof is the same!