. Suppose 10 i.i.d. observations result in ¯ x = 47. Let the likelihood for μ correspond to the model ¯ X | μ ∼ N ( μ, 50 / 10), and the prior for ( μ − 50) / 8 be Student’s t with 1 degree of...

.
Suppose 10 i.i.d. observations result in ¯x
= 47. Let the likelihood for
μ
correspond to the model ¯X
|
μ
∼
N(μ,
50/10), and the prior for (μ
− 50)/8 be Student’s
t
with 1 degree of freedom.

a.
Show that the five-point Gauss–Hermite quadrature rule relies on the Hermite polynomial
H5(x) =
c(x5 − 10x3 + 15x).

b.
Show that the normalization of
H5(x) [namely,
H5(x),H5(x)_ = 1] requires
c
= 1/
2π. You may wish to recall that a standard normal distribution has odd moments equal to zero and
rth moments equal to
r!/[(r/2)!2r/2] when
r
is even.

c.
Using your favorite root finder, estimate the nodes of the five-point Gauss–Hermite quadrature rule. (Recall that finding a root of
f
is equivalent to finding a local minimum of |f
|.) Plot
H5(x) from −3 to 3 and indicate the roots.

d.
Find the quadrature weights. Plot the weights versus the nodes. You may appreciate knowing that the normalizing constant for
H6(x) is 1/

e.
Using the nodes and weights found above for five-point Gauss–Hermite integration, estimate the posterior variance of
μ. (Remember to account for the normalizing constant in the posterior before taking posterior expectations.)