For a < b, the right-triangular distribution has density functionand the left-triangula...

For a < b, the right-triangular distribution has density function

and the left-triangular distribution has density function

These distributions are denoted by RT(a, b) and LT(a, b), respectively.

(a) Show that if X ~ RT(0, 1), then Xʹ = a + (b – a)X ~ RT(a, b); verify the same relation between LT(0, 1) and LT(a, b). Thus it is sufficient to generate from RT(0, 1) and LT(0, 1).

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(b) Show that if X ~ RT(0, 1), then 1 – X ~ LT(0, 1). Thus it is enough to restrict our attention further to generating from RT(0, 1).

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(c) Derive the inverse-transform algorithm for generating from RT(0, l). Despite the result in (b), also derive the inverse-transform algorithm for generating directly from LT(0, l).

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(d) As an alternative to the inverse-transform method, show that if U₁ and U₂ are IID U(0, 1) random variables, then max{U₁, U₂} ~ RT(0, 1). Do you think that this is better than the inverse-transform method? In what sense? (See Example 8.4.)

Example 8.4