Univariate normal distribution:
Let X = (X_{I}, X_{2}, ..., X_{n},) has the multivariate normal distribution (5.26) of Section 5.4. Show that Y = a X ' follows a univariate normal distribution where a = ( a_{1} a_{2} ..., a_{n}, ). Also find the mean and the variance of Y.
Solution:
We have
M_{y}(t)= E (e^{tY})= E(e^{taX'}) = E(e^{τX'})
where τ = ta = (ta_{1}, ta_{2},..., ta_{n} ).
From Example 3 of Section 5.4, we have
M_{X} (t) = E(e ^{tX'}) = e ^{τμ' +1/2(t∑ t')}
Hence
M_{y}(t) = M_{X}(τ) = e^{τμ' - ½ (τ∑τ') }=e ^{t(a μ')-1/2 t2(a∑a')}
=e ^{tμ} -1/2 t^{2} σ^{2}
Where
Hence
M_{X}(t) = e ^{tμ+1/2 (t2σ2)}
in other words Y - N ( μ, σ^{2} ) where