TOPIC:Uniform distribution.
Let U and V be independent Uniform(0, 1) random variables. (a) Calculate E(Uk) where k> 0...
Let U and V be independent Uniform[0, 1] random variables. (a) Calculate E(Uk) where k > 0 is some fixed constant (b) Calculate E(VU)
Let U and V be independent Uniform(0, 1) random variables. (a) Calculate E(Uk) where k > 0 is some fixed constant. (b) Calculate E(VU).
Let U., Un be independent, identically distributed Uniform random variables with (continu- ous) support on (0, b), where b >0 is a parameter. Define the random variable Y :--Σίι log(U), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b of Y by explicitly computing it.
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
Help please! Let Be be Brownian motion and fix to > 0. Prove that By: = Bto+t - Blo; t o is a Brownian motion.
PROB5 Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
7. Suppose the random variable U has uniform distribution on [0,1]. Then a second random variable T is chosen to have uniform distribution on [O, U] Calculate P(T > 1/2)
n. 7. Let Xi, , Xn be iid ;0) =-e-r2/0 where x > 0. Sho w that θ=「x? is based on f (x efficient.