STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 2 Tutorial
1. If X is a uniform random variable on [0, 1] show that U = − log(X) is an exponential random variable. Find E(−log(X)) directly and verify the result by finding E(U) from the density of U.
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2. Find the frequency function of Y = X2 if X has frequency function
(a) f(x) = 2xe−x2, x ≥ 0; (b) f(x)=e−x, x≥0;
(c)f(x)=12, −1≤x≤1.
3. The random variable X has p.d.f.
3×2 f(x) = 2 ,
Find the p.d.f. of Y = |X|.
4. Let X = (X, Y ) be a random vector with joint density function f(x,y)=2x+2y−4xy, 0≤x≤1, 0≤y≤1.
(a) Determine the marginal densities of X and Y . (b) Find the mean and variance of X.
(c) Calculate E(XY ).
(d) Find the covariance of X and Y .
(e) Find E(X|Y = 41).
5. Find the distributions of Y1 (minimum) and Yn (maximum) for random samples of size n from a population having the Beta distribution with α = 3, β = 2. The density function of Beta(α, β) is given by
f(x;α,β) = Γ(α+β)xα−1(1−x)β−1 for0