Zero-inflated distributions

Zero-inflated distributions are used to model count data that have many zero counts. For example, a zero-inflated Poisson distribution is a random variable X takes the value 0 with probability . Otherwise, with probability 1, it follows a Poisson distribution with mean . Thus, the mass function of X can be written as: fX(x)=I0(x)+(1)fY(x),x=0,1,2, where I0() is an indicator function (taking value 1 when x equals to 0 and value 0 otherwise) and fY() is the mass function of a Poisson distribution with parameter . (a) Verify that fX is a mass function. (b) Find the mean and variance of X in terms of and . Zero-inflated distributions are used to model count data that have many zero counts. For example, a zero-inflated Poisson distribution is a random variable X takes the value 0 with probability . Otherwise, with probability 1, it follows a Poisson distribution with mean . Thus, the mass function of X can be written as: fX(x)=I0(x)+(1)fY(x),x=0,1,2, where I0() is an indicator function (taking value 1 when x equals to 0 and value 0 otherwise) and fY() is the mass function of a Poisson distribution with parameter . (a) Verify that fX is a mass function. (b) Find the mean and variance of X in terms of and