6. Consider the pdf of the Uniform distribution. 5(8:2,5) = {B=A 1 A<x<B f(x; A,B) =...
Suppose X has the following Uniform distribution if 0<x<6 f(x)=\ & 0 otherwise a) Sketch the pdf of X b) What is Pr(X<4)? c) What is Pr(X<2|X<4)?
СТ 5. The triangular distribution has pdf 0<<1 f(x) = (2-2) 1<x<2. It is the sum of two independent uniform(0.1) random variables. (a) Find c so that f(x) is a density function. (b) Draw the pdf, and derive the cdf using simple geometry. (c) Derive the cdf from its definition. (d) Derive the mean and variance of a random variable with this distribution.
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
(30%) X has the uniform pdf f, (x)- b-a otherwise a) Determine the Probability Distribution Function F(). b) Determine ECx), E(x2) ando,
A continuous random variable X has a beta distribution with
p.d.f :
1 f(x) = 0<<<1, a > 2 B(4, 5)22-1(1 – 2)8-1 Determine E (3) HINT: E possible. (-) + E(X) Please show your work and simplify your final answer as much as
please show your work
Part 1. The Continuous Uniform Distribution 5. The pdf for this In this part of the lab, we'll work with the continuous uniform distribution on the interval (2.5) - in other words, min = 2 and max distribution is flz) 2<x<5 10, otherwise Compute the following by hande, using pencil and paper): (a) P(XS4) b) P(X > 4) (c) P(3 < X 4.5) (d) Find the median of this distribution, ie, the value of such that...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
pectively, 3. Ajoint pdf is defined by (C(x + 2y), for 0 <x< 2, and 0 < y< 1, fx.r(x,y) = -{-4** 0. otherwise. a. Find the value of C. b. Find the marginal pdf of X alone. 9 c. Find the pdf of U = U = x+132 4. Consider n independent rvs XX2, ..., X, having the same distribution with a common variance a?. For any i = 1,2, ..., n, find Cov(x,- 8, 8), where 8 =...
3. Consider a continuous random variable X with pdf given by 0, otherwise This is called the exponential distribution with parameter X. (a) Sketch the pdf and show that this is a true pdf by verifying that it integrates to 1 (b) Find P(X < 1) for λ (c) Find P(X > 1.7) for λ : 1
4. (6 marks) Consider a random sample of size n from a distribution with pdf f(x:0) 26-1 if 0 1 and zero otherwise; θ 0, Find the UMVUE of 1/θ x