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Just a note

Do not confuse the gamma function with the gamma distribution. The gamma function is closely related to factorials, except that factorials only work with whole numbers (of zero and above) - whereas the gamma function can be applied to any real number above zero.

Algebraically speaking the gamma function and factorials are related as follows:

gamma(x) = factorial(x-1) and gamma(x-1) = factorial(x) and factorial(x) = x*factorial(x-1) and gamma(x) = x*gamma(x-1)

More straightforwardly the factorial of some number x, is often abbreviated to x!, where x! = x × (x-1) × (x-2) × (x-3) ... × 1.

Thus 4! = 4×3×2×1 = 18 and Γ(4) = 3! = 6.

Notice that, by convention 0! = Γ(1) = 1 = 1! = Γ(2)