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Linear Combinations of Independent Random Variables

Learn about linear combinations, chi-square distribution, examples, and important facts regarding independent and normal random variables. Explore proof and claims in this informative week 2 study material.

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Linear Combinations of Independent Random Variables

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  1. Linear Combination of Two Random Variables • Let X1 and X2 be random variables with • Let Y = aX1 + bX2. Then,… week2

  2. Linear Combination of Independent R.V • Let X1, X2,…, Xn be independent random variables with • Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,… week2

  3. Linear Combination of Normal R.V • Let X1, X2,…, Xn be independent normally distributed random variables with • Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,… week2

  4. Examples week2

  5. The Chi Square distribution • If Z ~ N(0,1) then, X = Z2 has a Chi-Square distribution with parameter 1, i.e., • Can proof this using change of variable theorem for univariate random variables. • The moment generating function of X is week2

  6. Important Facts • If , all independent then, • If Z1, Z2,…Zn are i.i.d N(0,1) random variables. Then, week2

  7. Claim • Suppose X1, X2,…Xnare i.i.d normal random variables with mean μ and variance σ2. Then, are independent standard normal variables, where i = 1, 2, …, n and week2

  8. Important facts • Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ and variance σ2. Then, Proof:… • This implies that… • Further, it can be shown that and s2 are independent. week2

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