Calculate P(X(t)=x), the probability that a randomly selected customer makes exactly x transactions in the interval (0, t].

pnbd_nocov_PMF(r, alpha_0, s, beta_0, x, vT_i)

pnbd_staticcov_PMF(r, s, x, vAlpha_i, vBeta_i, vT_i)

## Arguments

r

shape parameter of the Gamma distribution of the purchase process. The smaller r, the stronger the heterogeneity of the purchase process

alpha_0

rate parameter of the Gamma distribution of the purchase process

s

shape parameter of the Gamma distribution for the lifetime process. The smaller s, the stronger the heterogeneity of customer lifetimes

beta_0

rate parameter for the Gamma distribution for the lifetime process.

x

The number of transactions to calculate the probability for (unsigned integer).

vT_i

Number of periods since the customer came alive.

vAlpha_i

Vector of individual parameters alpha.

vBeta_i

Vector of individual parameters beta.

## Value

Returns a vector of probabilities.

## References

Schmittlein DC, Morrison DG, Colombo R (1987). “Counting Your Customers: Who-Are They and What Will They Do Next?” Management Science, 33(1), 1-24.

Bachmann P, Meierer M, Naef, J (2021). “The Role of Time-Varying Contextual Factors in Latent Attrition Models for Customer Base Analysis” Marketing Science 40(4). 783-809.

Fader PS, Hardie BGS (2005). “A Note on Deriving the Pareto/NBD Model and Related Expressions.” URL http://www.brucehardie.com/notes/009/pareto_nbd_derivations_2005-11-05.pdf.

Fader PS, Hardie BGS (2007). “Incorporating time-invariant covariates into the Pareto/NBD and BG/NBD models.” URL http://www.brucehardie.com/notes/019/time_invariant_covariates.pdf.

Fader PS, Hardie BGS (2020). “Deriving an Expression for P(X(t)=x) Under the Pareto/NBD Model.” URL https://www.brucehardie.com/notes/012/pareto_NBD_pmf_derivation_rev.pdf