3. If U1 and U2 are independent standard uniform random variables, show that the variables are...
1 (10pts) Let U1, U2, ... ,Un be independent uniform random variables over [0, 0] with the probability density function (p.d.f). () = a 2 + [0, 03, 0 > 0. Let U(1), U(2), .-. ,U(n) be the order statistics. Also let X = U(1)/U(n) and Y = U(n)- (a) (5pts) Find the joint probability density function of (X, Y). (b) (5pts) From part (a), show that X and Y are independent variables.
(10 points) Consider the infinite sequence of independent and identically distributed (stan- dard) uniform random variables: U1, U2, ..., i.e., Ui » Uniform(0,1). Also let N ~ Poisson(a). Assume N is independent of {U;}i>1. Consider the random variable z = į V. Calculate EZ. (Hint: use conditioning.)
Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0, 1]. Let Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0,...
3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and Xn+1-Xn +2Un-1, 1,2,.. (a) Show that X, n 0,1,2, is a Markov chain, and give its transition graph. (b) Find EX and Var(X) c)Give P(X
Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0<u < 1 !-un . 1- e-". (a) P(Ui-u and N = n) = (b) PUI S u and N is even) Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
1. Let Y.Y2, ,y, be independent and identically distributed N(μ, σ2) random variables. Show that, where d() denotes the cumulative distribution function of standard normal [You need to show both the equalities]
, Yn be independent and identically distributed N(μ, σ2) random variables. Show Let YİM, that, where φ(-) denotes the cumulative distribution function of standard normal. [You need to show both the equalities]
5. (10 marks) (a) Let E, E2} be mutually independent random variables. Show that the conditional density T(e1, e2 x) can be written in the form (4 marks) (b) Let a set of observations Y be of the form Yk exp(r)Ek , k = 1,... , M where ER.Let Ek be mutually independent and identically distributed and normal with T(ek) N(He,©?) for all k (i) Derive the likelihood density n(y|x) (ii) Derive the maximum likelihood estimate ML (3 marks) (3...
1. Let Yi,Y2, ,y, be independent and identically distributed N( 1,02) random variables. Show that, EVn P( Y where ) denotes the cumulative distribution function of standard normal You need to show both the equalities