expected number of counts that have The variance of Y equals p. The equality occurred: for the distribution this will also of the mean and the variance is known as be the mean thus count variables have If y=0, then Pr(r=0)=exp(u) greater than the mean, which is called overdispersion. Sometimes, therefore I y=l, then Pr(r=1)=exp(u )u the Poisson regression model is not If y=2, then Pr(=2)=exp(-u )u/2 entirely appropriate, often leading the analyst to the negative binomial regression If y=3, then Pr(r=3)=exp(u )u/6 model( to be discussed later) Some properties of a As g increases, the probability of Os Poisson distribution decreases. In a poisson distribution for F=0.8, the probability of an 0 is 0.45; As u increases the mass of the for p=1.5, the probability of an 0 is 0.22; distribution shifts to the right; we'llsee for !=2.9, the probability of an 0 is 0.05, this below in the sample graphs of the for A=10.5, the probability of an 0 is univariate Poisson distribution As u increases the Poisson distribution approximates a normal distribution5 9 • where the parameter μ represents the expected number of counts that have occurred; for the distribution this will also be the mean. Thus, 10 Some properties of a Poisson distribution • As μ increases, the mass of the distribution shifts to the right; we’ll see this below in the sample graphs of the univariate Poisson distribution. 6 11 • The variance of Y equals μ. The equality of the mean and the variance is known as equidispersion. Actually in practice, many count variables have a variance greater than the mean, which is called overdispersion. Sometimes, therefore, the Poisson regression model is not entirely appropriate, often leading the analyst to the negative binomial regression model (to be discussed later). 12 • As μ increases, the probability of 0’s decreases. In a Poisson distribution, for μ = 0.8, the probability of an 0 is 0.45; for μ = 1.5, the probability of an 0 is 0.22; for μ = 2.9, the probability of an 0 is 0.05; for μ = 10.5, the probability of an 0 is 0.00002. • As μ increases, the Poisson distribution approximates a normal distribution