4) Suppose that Y~exp(8). Let X = ln(Y). Find the pdf of X. 5) Let Y and Y2 be iid U(0,1). Let S YY2. Find the pdf of S.
Suppose that Y1,Y2,··· ,Yn is an iid from Y ∼ U(0,3). Find the limiting distribution of ¯ Y . What is the probability of average of Y from a random sample of 10 that exceed 1.6?
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...
Let X1, . . . , Xn ∼ iid Exp(λ) and Y1, . . . , Ym ∼ iid Exp(τ ) be independent random samples. (a) Find the restricted MLEs under the null hypothesis H0 : λ = τ . (b) Write out a formula for the LRT statistic, and describe how you could perform this test asymptotically.
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 = X(1), and Yk = X(k) – X(k-1), 2<k<n. Find the joint pdf of Yı,Y2, ...,Yn. Hint: Note that (Y1,Y2, ...,Yn) = g(X(1), X(2), ..., X(n)), where g is invertible and differentiable. Use the change of variable formula to derive the joint pdf of Y1, Y2, ...,Yn.
Suppose Y1 and Y2 are independent normal with same variance. (a) Show that U1 = Y1 +Y2 and U2 = Y1 - Y2 are joint normal. (b) Show that U1 = Y1 +Y2 and U2 = Y1 - Y2 are independent.
X1 and X2 are IID with the density: a) joint density with r.v. Y1 = X1 and Y2 = X1 + X2 (I think this might be transformation but I'm stuck after) b) marginal density for Y2 f(x) = V2nz exp (-2) , for x>0,
Suppose that a random sample of size 36, Y1,Y2,...,Y36, is drawn from a uniform pdf de ned over the interval (0, θ), where θ is unknown. Set up a large- sample sign test for deciding whether or not the 25th per- centile of the Y -distribution is equal to 6. Let α = 0.05. With what probability will your procedure commit a Type II error if 7 is the true 25th percentile?