Linear Combination of Two Random Variables

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

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

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

Examples week2

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

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

Claim Suppose X1, X2,…Xn are 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

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