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The distribution of the sum of r-th power of standard normal random variablesis a generalization of the chi-squared distribution. In this paper, werepresent the probability density function of the random variable by anone-dimensional absolutely convergent integral with the characteristicfunction. Our integral formula is expected to be applied for evaluation of thedensity function. Our integral formula is based on the inversion formula, andwe utilize a summation method. We also discuss on our formula in the view pointof hyperfunctions.

An integral formula for the powered sum of the independent, identically and normally distributed random variables