Truncated Normal Density. Let x be N(1, 1). Using the results in the Appendix, show that: (a) The conditional density f(x/x > 1) = 2(x

Truncated Normal Density. Let x be N(1, 1). Using the results in the Appendix, show that:

(a) The conditional density f(x/x > 1) = 2φ(x − 1) for x > 1 and f(x/x < 1) = 2φ(x − 1) for x < 1.

(b) The conditional mean E(x/x > 1) = 1 + 2φ(0) and E(x/x < 1) = 1 − 2φ(0). Compare with the unconditional mean of x.

(c) The conditional variance var(x/x > 1) = var(x/x < 1) = 1 − 4φ2(0). Compare with the unconditional variance of x.