{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe use the general formula for the variance of a sum and show that all covariances vanish.\n\n1. Let $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $E[X_i^2] \u003c \\infty$ for each $i$.\n2. Fix $i \u003c j$. Apply the definition of independence to the set $J = \\set{i, j}$. For all Borel sets $A$ and $B$, it gives $P(X_i \\in A, X_j \\in B) = P(X_i \\in A) \\, P(X_j \\in B)$.\n3. So $X_i$ and $X_j$ are [[independent as a pair|0E9502NT92]].\n4. Both have finite second moments. Since [[independent random variables are uncorrelated|077GN9HDR6]], $\\operatorname{Cov}(X_i, X_j) = 0$.\n5. By the [[formula for the variance of a sum|0D86GJEZ39]],\n$$\\operatorname{Var}(X_1 + \\dots + X_n) = \\sum_{i=1}^n \\operatorname{Var}(X_i) + 2 \\sum_{1 \\le i \u003c j \\le n} \\operatorname{Cov}(X_i, X_j).$$\n6. By step 4, every covariance term is $0$. So $\\operatorname{Var}(X_1 + \\dots + X_n) = \\operatorname{Var}(X_1) + \\dots + \\operatorname{Var}(X_n)$.\n\n## Idea\n\nThe variance of a sum is the sum of the variances plus twice the sum of all covariances. Independence of each pair kills every covariance, so only the variances remain.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "60A05"
    ],
    "name": "Proof of the formula for the variance of a sum of independent random variables",
    "proves": "0V0778DCW9",
    "sources": [
      {
        "locator": "Sum of uncorrelated variables",
        "url": "https://en.wikipedia.org/wiki/Variance"
      }
    ],
    "tag": "TCV030YM49",
    "uses": [
      "0XCE070V8B",
      "0E9502NT92",
      "0D86GJEZ39",
      "077GN9HDR6"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/TCV030YM49/",
  "links": [
    {
      "field": "proves",
      "target": "0V0778DCW9"
    },
    {
      "field": "uses",
      "target": "0XCE070V8B"
    },
    {
      "field": "uses",
      "target": "0E9502NT92"
    },
    {
      "field": "uses",
      "target": "0D86GJEZ39"
    },
    {
      "field": "uses",
      "target": "077GN9HDR6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/TCV030YM49.md",
  "msc": [
    "60A05"
  ],
  "name": "Proof of the formula for the variance of a sum of independent random variables",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TCV/TCV030YM49.md",
  "status": null,
  "tag": "TCV030YM49",
  "uri": "m2://TCV030YM49",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-formula-for-the-variance-of-a-sum-of-independent-random-variables/",
  "writing": {
    "tag": "TCV030YM49",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}