{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe compute the characteristic function of $X + Y$ and recognise it.\n\n1. For a real random variable $Z$, let $\\varphi_Z(t) = E[e^{itZ}]$ be its [[characteristic function|0MYDNRT7PP]], for $t \\in \\R$.\n2. By the [[characteristic function of the normal distribution|0NS0TDA4E2]], $\\varphi_X(t) = \\exp\\left(i \\mu_1 t - \\tfrac12 \\sigma_1^2 t^2\\right)$ and $\\varphi_Y(t) = \\exp\\left(i \\mu_2 t - \\tfrac12 \\sigma_2^2 t^2\\right)$.\n3. $X$ and $Y$ are [[independent|0E9502NT92]]. By the [[characteristic function of a sum of independent random variables|02JGYG0T7M]], $\\varphi_{X+Y}(t) = \\varphi_X(t) \\, \\varphi_Y(t)$ for each $t$.\n4. By the [[product of exponentials|0WM7DJCM6G]], $\\varphi_{X+Y}(t) = \\exp\\left(i (\\mu_1 + \\mu_2) t - \\tfrac12 (\\sigma_1^2 + \\sigma_2^2) t^2\\right)$.\n5. Let $\\mu = \\mu_1 + \\mu_2$ and $\\sigma^2 = \\sigma_1^2 + \\sigma_2^2$. Then $\\sigma^2 \u003e 0$, so the [[normal distribution|07RXBVTVZB]] $\\mathcal{N}(\\mu, \\sigma^2)$ is defined. Let $W \\sim \\mathcal{N}(\\mu, \\sigma^2)$.\n6. By the characteristic function of the normal distribution, $\\varphi_W(t) = \\exp\\left(i \\mu t - \\tfrac12 \\sigma^2 t^2\\right)$. By step 4, $\\varphi_W(t) = \\varphi_{X+Y}(t)$ for all $t \\in \\R$.\n7. By the theorem that [[the characteristic function determines the distribution|0T8BYAFEXX]], $X + Y$ and $W$ have the same law.\n8. Hence $X + Y \\sim \\mathcal{N}(\\mu_1 + \\mu_2, \\, \\sigma_1^2 + \\sigma_2^2)$.\n\n## Idea\n\nFor independent variables, characteristic functions multiply. The characteristic function of a normal law is the exponential of a quadratic in $t$. So in a product the means add and the variances add. Since a characteristic function fixes the law, the sum is normal.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "kind": "proof",
    "msc": [
      "60E05"
    ],
    "name": "Proof of the sum of independent normal random variables by characteristic functions",
    "proves": "0NXGPJWNFN",
    "sources": [
      {
        "locator": "Proof using characteristic functions",
        "url": "https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables"
      }
    ],
    "tag": "G7Q5GT80HP",
    "uses": [
      "07RXBVTVZB",
      "0MYDNRT7PP",
      "0NS0TDA4E2",
      "02JGYG0T7M",
      "0T8BYAFEXX",
      "0E9502NT92",
      "0WM7DJCM6G"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/G7Q5GT80HP/",
  "links": [
    {
      "field": "proves",
      "target": "0NXGPJWNFN"
    },
    {
      "field": "uses",
      "target": "07RXBVTVZB"
    },
    {
      "field": "uses",
      "target": "0MYDNRT7PP"
    },
    {
      "field": "uses",
      "target": "0NS0TDA4E2"
    },
    {
      "field": "uses",
      "target": "02JGYG0T7M"
    },
    {
      "field": "uses",
      "target": "0T8BYAFEXX"
    },
    {
      "field": "uses",
      "target": "0E9502NT92"
    },
    {
      "field": "uses",
      "target": "0WM7DJCM6G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/G7Q5GT80HP.md",
  "msc": [
    "60E05"
  ],
  "name": "Proof of the sum of independent normal random variables by characteristic functions",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/G7Q/G7Q5GT80HP.md",
  "status": null,
  "tag": "G7Q5GT80HP",
  "uri": "m2://G7Q5GT80HP",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-sum-of-independent-normal-random-variables-by-characteristic-functions/",
  "writing": {
    "tag": "G7Q5GT80HP",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}