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Univariate normal distribution:
Let X = (XI, X2, ..., Xn,) has the multivariate normal distribution (5.26) of Section 5.4. Show that Y = a X ' follows a univariate normal distribution where a = ( a1 a2 ..., an, ). Also find the mean and the variance of Y.
Solution:
We have
My(t)= E (etY)= E(etaX') = E(eτX')
where τ = ta = (ta1, ta2,..., tan ).
From Example 3 of Section 5.4, we have
MX (t) = E(e tX') = e τμ' +1/2(t∑ t')
Hence
My(t) = MX(τ) = eτμ' - ½ (τ∑τ') =e t(a μ')-1/2 t2(a∑a')
=e tμ -1/2 t2 σ2
Where
MX(t) = e tμ+1/2 (t2σ2)
in other words Y - N ( μ, σ2 ) where
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