2. Let A4, be the discrete uniforin on (1/20,2/29,3/2, in distribution to a Uniform(0,1). o 1)/,...
Let X1, . . . , Xn be a random sample from the discrete uniform distribution on 1, 2, . . . , θ. Using the definition of sufficient statistic, show that X(n) is a sufficient statistic for θ.
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].
Suppose that X1,X2,... is a sequence of i.i.d. r.v.s having uniform distribution on [0,1]. Define Yn=n(1−max1≤i≤nXi) for n=1,2,.... Prove that Yn converges in distribution to an exponential distribution.
a. Let X ~ Uniform(0,1). Find the distribution function of Y =-21nX. What is the distribution of Y. Find P(Y> 0.01)
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S. 6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let Calculate analytically the variance of X. (HINT: E g(x)f(x)dx, and the p.d.f. 10SzSI 0 o.t.w. f(x) of a uniform distribution is f(x) =
1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence class such that x ~ y (x is equivalent to y) if x-y є Q, the set of rational numbers 2. Given 1, by the Axiom of Choice, there exists a nonempty set B C [0,1) such that IB contains exactly one member of each equivalence class. Prove each of the following (a) Suppose that q E Qn [0, 1). Show that B (b)...
7.103 Let X have the discrete uniform distribution on -1,0, 1) and let Y X a) Show that ρ(X, Y) = 0, although X and Y are functionally related and hence associated. b) Why doesn't the result of part (a) conflict with the correlation coefficient's role as a measure of association?
prove that A is non singular 5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?