Let Y have a truncated distribution with pdf g(y) (y) (b)(a) , for a y b, zero elsewhere, where (x) and (x) are, respectively, the pdf and distribution function of a standard normal distribution Show then that E(Y) is equal to (a) (b) (b) (a) ...

The expected value of a truncated random variable is defined as EY bafyydy ...

Let Y have a truncated distribution with pdf g(y) = (y)/[(b)(a)], for a < y < b,

Question:

Let Y have a truncated distribution with pdf g(y) = φ(y)/[Φ(b)−Φ(a)], for a < y < b, zero elsewhere, where φ(x) and Φ(x) are, respectively, the pdf and distribution function of a standard normal distribution. Show then that E(Y) is equal to [φ(a) − φ(b)]/[Φ(b) − Φ(a)].

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