{
  "id": "2408.04101",
  "version": "v1",
  "published": "2024-08-07T21:37:59.000Z",
  "updated": "2024-08-07T21:37:59.000Z",
  "title": "On the product of correlated normal random variables and the noncentral chi-square difference distribution",
  "authors": [
    "Robert E. Gaunt"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as the noncentral chi-square difference distribution). As a consequence, we obtain, amongst other results, an exact formula for the probability density function of the noncentral chi-square difference distribution, a Stein characterisation of the noncentral chi-square difference distribution, a simple formula for the moments of the sum of independent copies of the product of correlated normal random variables and an exact formula for the probability that such a random variable is negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T21:37:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "62E15"
    ],
    "keywords": [
      "noncentral chi-square difference distribution",
      "correlated normal random variables",
      "independent noncentral chi-square random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}