Evaluating the Probability Integral Using Wallis's Product Formula for π

By “straightforward” we mean that it can be presented in a standard one-variable calculus course. This is not the first time an “elementary” proof of this identity has been given. Several other proofs can be found in, e.g., [1], [3], [4], [5], [7]. Most of them are variations on a theme (double integral, gamma function, . . . ). This one is a simplification of one of the most recent ones by Lord [6]. The method is based on the following lemma which we will prove later: Lemma. The function F with F(x) = e−x − (1 − x n )n satisfies: