Let U., Un be independent, identically distributed Uniform random variables with (continu- ous) support on (0,...
1. [26 pts Let Uı, , Un be independent, identically distributed Unifomn random variables with (continu- ous) support on (0, b), where b> 0 is a parameter. (a) Define the random variable Y :--Σί 1 log(U,), where log is the natural logarithm function. De- termine the probability density function (pdf) p(y; b) ofY by explicitly computing it (b) Based on the pdf you found in part (a) above, determine the third moment of Y, i.e., EY] (c) Suppose now that...
4.4-14. Let X and Y be random variables of the continu- ous type having the joint pdf f(x,y) = 8xy, 0<xsys 1. Draw a graph that illustrates the domain of this pdf. (a) Find the marginal pdfs of X and Y. (b) Compute jx, My, o, oz, Cov(X,Y), and p. (c) Determine the equation of the least squares regres- sion line and draw it on your graph. Does the line make sense to you intuitively?
Question 3 15 marks] Let X1,..,X be independent identically distributed random variables with pdf common ) = { (#)%2-1/64 0 fx (a;e) 0 where 0 >0 is an unknown parameter X-1. Show that Y ~ T (}, ); (a) Let Y (b) Show that 1 T n =1 is an unbiased estimator of 0-1 ewhere / (0; X) is the log- likeliho od function; (c) Compute U - (d) What functions T (0) have unbiased estimators that attain the relevant...
If X and Y are independent and identically distributed uniform random variables on (0,1) compute the joint density of U = X+Y, V = X/(X+Y) Part A, The state space of (U,V) i.e. the domain D over which fU,Y (u,v) is non-zero can be expressed as (D = {(u,v) R x R] 0 < h1(u,v) < 1, 0 < h2(u,v) < 1} where x = h1 (u,v) and y = h2 (u,v) Find h1(u,v) = (write a function in terms...
Let X and Y be independent and identically distributed with marginal probability density function f(a)- 0 otherwise, where 8>0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y. (b) [6 pts) Find the joint probability density function of U X+Y and V- X+Y
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.
The random variables X1, X2, - .. are independent and identically distributed with common pdf 0 х > fx (x;0) (2) ; х<0. This distribution has many applications in engineering, and is known as the Rayleigh distribution. 2 (a) Show that if X has pdf given by (2), then Y = X2/0 is x2, i.e. T (1, 2) i.e. exponential with mean 2, with pdf fr (y;0) - ; y0; (b) Show that the maximum likelihood estimator of 0 is...
(1 point) If X and Y are independent and identically distributed uniform random variables on (0, 1), compute each of the following joint densities. (a) U -3X, V - 3X/Y. fu.v(u, v) - (b) U - 5X + Y, V - 3X/(X + Y)
2. [12 marksj Let Xi and X2 be independent and identically distributed random variables, each having an exponential distribution with density function (x),foro, 0, elsewbere Pdof W Let W = X1 +X2 and's Use the -method-of transformatiou- to find jhe joint probability density fuactíion of-W andy. AreWandfindependent?AThy? M covered m w, r 201 Instead tyto ind pdf of w b methed of colf
5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6 pts] Use the convolution formula to find the probability density function of X +Y (b) (6 pts) Find the joint probability density function of V= X + Y U=X+Y and 5. Let X and Y be independent and identically distributed with marginal probability density function İf a> 0, otherwise, e-ea f(a)-( where >0 (a) [6...