Univariate normal distribution:

Let X = (X_I, X₂, ..., 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 = ( a₁ a₂ ..., 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 = (ta₁, ta₂,..., 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 t² σ²

Where

Hence

M_X(t) = e ^{tμ+1/2 (t2σ2)}

in other words Y - N ( μ, σ² ) where