  The initial mass function (IMF) for the stars is usually fitted by three
straight lines, which means seven parameters. The presence of brown dwarfs (BD)
increases to four the straight lines and to nine the parameters. Another common
fitting function is the lognormal distribution, which is characterized by two
parameters. This paper is devoted to demonstrating the advantage of introducing
a left truncated beta probability density function, which is characterized by
four parameters. The constant of normalization, the mean, the mode and the
distribution function are calculated for the left truncated beta distribution.
The normal-beta (NB) distribution which results from convolving independent
normally distributed and beta distributed components is also derived. The
chi-square test and the K-S test are performed on a first sample of stars and
BDs which belongs to the massive young cluster NGC 6611 and on a second sample
which represents the star's masses of the cluster NGC 2362.
