{'fraction_non_alphanumeric': 0.02912621359223301, 'fraction_numerical': 0.004160887656033287, 'mean_word_length': 4.68503937007874, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Contributions to the Theory of the Non-Central X2 Distribution

In this paper, we consider the probability density function (pdf) of a non-central χ distribution with odd number of degrees of freedom ν. This pdf is represented in the literature as an infinite sum. Consequently, we present three alternative expressions to this pdf. The first expression is in terms of a partial derivative of the hyperbolic cosine function. The second expression, on the other hand, is a finite sum representation of (ν + 1)/2 terms only instead of the infinite sum. Finally, we present a general recurrence relation for such pdf. These results have applications in approximation of the pdf of non-central χ distributed random variables.