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and Poisson Exponential, Connected The Poisson distribution counts the number of discrete events in a fixed time period; it is closely connected to the exponential distribution, which (among other applications) measures the time between arrivals of the events. The Poisson distribution is a discrete distribution; the random variable can only take nonnegative integer values. The exponential distribution can take any (nonnegative) real value. The Poisson process is completely described by a parameter , the average number of arrivals per unit time. If X denotes the number of events which happen in unit time, then k - e . k! This is connected to the random variable Y , the time between successive events, by the fact that Y is exponentially distributed with density function e-x , and therefore P (X = k) =
t
(1) (2)
P (Y t) =
0
e-x dx = 1 - e-t ,
P (Y > t) = 1 - P (Y t) = e-t .
Sometimes we may wish to speak about the number of arrivals in a time interval T which is different from the unit time interval. This is done by scaling: multiply (average number of arrivals per unit time) by T (number of units of time) to get T , the average number of arrivals in time T . (So if an average of six babies arrive per day in a hospital, then an average of 0.25 babies arrive per hour.) The probability that exactly k events will occur in time interval T is therefore (3) (T )k -T e . k! This makes the connection between the Poisson and exponential distributions transparent: for the event Y > T means that the first arrival takes more than T units of time; which means that there are 0 arrivals in the first T units of time. Taking k = 0 in (3), this has probability e-T , i.e. P (Y > T ) = e-T . This was exactly what we claimed for the exponential distribution in (2). For an example, we will model the arrival of babies at a lying-in hospital. These are Poisson events, because they satisfy the three characteristics of a Poisson model: Arrivals of babies are independent events. (The mothers don't conspire together.) The probability of babies being born almost simultaneously is negligible. The probability of a baby being born in a time interval only depends on the length of the time interval, not on when it is during the day. For many years the hospital has been averaging 6 births per day. That means that the average time between births will be 1/6 of a day. Therefore if X denotes the number of babies born in a day, then 6k P (X = k) = e-6 , k!
1
2
while if Y denotes the time between births, then P (Y > t) = e-6t . The probability density function for Y is 6e-6x . Recall that the exponential which distribution has PDF e-x has mean 1/, so this fits: Y should have mean 1/6 (days/birth). Also note how easy it is to connect these parameters through "dimensional analysis": X, the number of births, is measured by babies/day; Y , the time between births, is measured in days/baby. So 6 babies/day is equivalent to 1/6 days/baby, or 4 hours/baby. For example, in the above scenario, given that a baby has just been born, the probability that the next baby will take one or more hours to arrive is found by converting hours to days, 1 hour = and computng P (Y > Surely time enough for a coffee break! Consistency of the Poisson model. How far can we trust the Poisson model? Actually, pretty far. We'll give an example to illustrate how consistent it is. It illustrates a more general fact (the sum of two independent Poisson RV's with mean and is a Poisson RV with mean + ). This is quite reasonable: think of two hospitals, one giving birth to an average of babies per day, the other giving birth to babies per day. If you combine the hospitals (say, a health conglomerate buys them both) then the conditions for a Poisson RV are satisfied for births at the combined hospitals, which would average + per day.) Consider a hospital which has an average of 6 babies born each day. There are two shifts for the nursing staff, so the administration prefers to use 12-hour periods, with an average of 3 babies born each 12-hour period. Does using different scales like this make a difference in the probabilities? No, if we keep the time intervals strai...

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