Data605-Week9-Discussion9-kamath

CHAPTER 9. CENTRAL LIMIT THEOREM

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Let X be a continuous random variable with mean μ(X) and variance σ^2(X), and let X^∗=(X−μ)σ be its standardized version. Verify directly that μ(X^∗)=0 and σ^2(X∗)=1.

\[X^∗= \frac{(X−μ)}{σ} \]

\[E(X^∗)=\frac{1}{σ}(E(X)−μ) \]

\[E(X^∗)=\frac{1}{σ}(μ−μ) \]

\[E(X^∗)=μ(X^∗)=0 \]

\[σ^2(X^∗)=E(\frac{X−μ}{σ})^2 \]

\[σ^2(X^∗)=\frac{1}{σ^2}σ^2 \]

\[σ^2(X^∗)=1 \]

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