Censored Normal Distribution. This is based on Greene (1993, pp. 692-693). Let y be N(, 2) and define y = y if y > c

Censored Normal Distribution. This is based on Greene (1993, pp. 692-693). Let y∗ be N(μ, σ2)

and define y = y∗ if y∗ > c and y = c if y∗ < c for some constant c.

(a) Verify the E(y) expression given in (A.7).

(b) Derive the var(y) expression given in (A.8). Hint: Use the fact that var(y) = E(conditional variance) + var(conditional mean)

and the formulas given in the Appendix for conditional and unconditional means of a truncated normal random variable.

(c) For the special case of c = 0, show that (A.7) simplifies to E(y) = Φ(μ/σ)

μ + σφ(μ/σ)

Φ(μ/σ)

and (A.8) simplifies to var(y) = σ2Φ

μ

σ

 

1 − δ

−μ

σ



+



−μ

σ

− φ(/σ)

Φ(μ/σ)

2

Φ



−μ

σ



where δ
−μ
σ

= φ(μ/σ)
Φ(μ/σ)
φ(μ/σ)
Φ(μ/σ)
+ μ
σ
. Similar expressions can be derived for censoring of the upper part rather than the lower part of the distribution.