1. Linear Combination of Two Random Variables
Let X1 and X2 be random variables with
Let Y = aX1 + bX2. Then,…
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2. Linear Combination of Independent R.V
Let X1, X2,…, Xn be independent random variables with
Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,…
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3. Linear Combination of Normal R.V
Let X1, X2,…, Xn be independent normally distributed random variables with
Let Y = a1X1 + a2X2 + ∙∙∙+ anXn. Then,…
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4. Examples
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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
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6. Important Facts If , all independent then,
If Z1, Z2,…Zn are i.i.d N(0,1) random variables. Then,
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7. 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
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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.
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