Let X1,X2,...,Xn be an independent and identically distributed (i.i.d.) random sample of Beta distribution with parameters...
Let X1,X2,...,Xn be an independent and identically distributed
(i.i.d.) random sample of Beta distribution with parameters α = 2
and β = 1, i.e., with probability density function fX(x) = 2x for x
∈ (0,1). Find the probability density function of the first and last
order statistics Y1 and Yn.
Homework Answers
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Let X1,X2,...,Xn be an independent and identically distributed
(i.i.d.) random sample of Beta distribution with parameters...
1. Let X1, X2,... , Xn be independent and identically distributed according to the unifornm distribution...
1. Let X1, X2,... , Xn be independent and identically distributed according to the unifornm distribution on (0,1). Let Xn and fn denote the 6th smallest and its pdf, respectively Determine fn(x) limn
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint pr...
Let (X1, Y1) and (X2, Y2) be independent and identically distributed continuous bivariate random variables with joint probability density function: fX,Y (x,y) = e-y, 0 <x<y< ; =0 , elsewhere. Evaluate P( X2>X1, Y2>Y1) + P (X2 <X1, Y2<Y1) .
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the...
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W)
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed,...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
Let X1, X2, ..., Xn be independent Exp(2) distributed random vari- ables, and set Y1 =...
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.
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval...
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval [0,1] ,Xn, each a) What is the pdf of Mmin(X1,X2, .. ,Xn)? b) Give the expectation and variance of XX 1-1лі.
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value...
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
1. Let X1, X2,... , Xn be independent and identically distributed according to the unifornm distribution on (0,1). Let Xn and fn denote the 6th smallest and its pdf, respectively Determine fn(x) limn
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W)
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
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.
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval [0,1] ,Xn, each a) What is the pdf of Mmin(X1,X2, .. ,Xn)? b) Give the expectation and variance of XX 1-1лі.
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