STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 2 Tutorial Solutions
1. Let X be a uniform random variable on [0, 1] and set U = − log(X).
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P(U ≤ u) = P(−log(X) ≤ u) = P(X≥e−u)
Thus U is an exponential rv with density fU (u) = e−u, u ≥ 0.
E(− log(X)) = E(U) =
− log(x).1dx = [x − x log(x)]0 = 1. ∞ −u
ue du = Γ(2) = 1.
2. IfY =X2 thenP(Y ≤y)=P(X2 ≤y)=P(−√y≤X≤√y).
(a) If X has density f(x) = 2xe−x2, x ≥ 0 then X takes positive values and Y has
f(√y)12y−1/2 = e−y, y ≥ 0. (b) If X has density f(x) = e−x, x ≥ 0 then Y has density
fY (y) = 12e−√yy−1/2, y ≥ 0.
(c) If X has density f (x) = 12 , −1 ≤ x ≤ 1 then Y has distribution function
FY (y) = P(−√y ≤ X ≤ √y) = 21(1 + √y) − 12(−√y + 1) = √y. Y hasdensityfY(y)=12y−1/2, 0≤y≤1.
3. NoteY =|X|so
P(Y ≤y)=P(−y≤X≤y)=y3, 0