SMp(x) Distribution Calculator

A probability distribution describes how probabilities are spread across the possible values of a random variable. For a continuous variable (like temperature), it assigns a probability density to each value, so the area under the curve over any interval equals the chance of the variable falling in that interval. For a discrete variable (like a count), it assigns a direct probability to each possible outcome.

The SMp(x) distribution is a six-parameter generalized probability function defined as SMp(x) = (x − a)^m · (b − x)^n · (c·x + d) for x in [a, b], and 0 otherwise. By choosing appropriate values of a, b, c, d, m, and n you can shape the distribution to closely approximate many standard distributions including the normal, beta, and Poisson distributions.

Yes. By setting parameters symmetrically (a and b equidistant from the mean, m = n, and appropriate c and d values), the SMp(x) function closely approximates the bell-curve shape of the normal distribution. It won't be identical since SMp(x) has finite support [a, b], but it can be a very good approximation over a chosen range.

Yes, with appropriate parameter choices. The Poisson distribution is discrete and right-skewed; by setting m small and n large (or vice versa) and adjusting c and d, the SMp(x) function can approximate the skewed shape of a Poisson distribution over a discrete range. The approximation improves as you fine-tune the six parameters.

The exponents m and n control the tail behavior at the lower bound a and upper bound b respectively. When m = n the distribution is symmetric; when m < n it is right-skewed; when m > n it is left-skewed. Both m and n must be positive for SMp(x) to be non-negative throughout [a, b].

Parameters c and d define the linear multiplier (c·x + d) in the SMp(x) formula. They shift and tilt the amplitude of the distribution across the domain. Setting c = 0 and d = 1 removes the linear tilt and gives a pure beta-like shape, while non-zero c values introduce an asymmetric amplitude gradient across x.

For SMp(x) to be a valid probability density function, it must be non-negative everywhere and integrate to 1 over [a, b]. Non-negativity requires m ≥ 0, n ≥ 0, and c·x + d ≥ 0 for all x in [a, b]. You then need to normalize by dividing by the integral. This calculator evaluates SMp(x) at your chosen x and flags whether the basic non-negativity condition is met.

Beyond normal and Poisson, the SMp(x) framework can approximate the beta, gamma, chi-squared, uniform, triangular, and exponential distributions, among others. The flexibility of six parameters makes it a versatile tool in statistical modeling, simulation design, and Bayesian prior specification.