2. Let Y = ex2, where X ~ Ņ(0, 1). (a) Find the p.d.f. of Y. J0 oPlY2 > ijdt. (b) C ompuite 2. Let Y = ex2, whe...
1. Let the joint p.d.f of X and Y be 2xe if 0 < x < 1 and y > x2 fxy(z, y) 0, otherwise. (a) Find the marginal p.d.f.'s of X and Y, respectively (b) Compute P(Y < 2X2) 1. Let the joint p.d.f of X and Y be 2xe if 0
4. Let X have p.d.f. fx(1),-1 < 2. Find the p.d.f. of Y-X2
Let (X, Y) be abivariate random vector with joint p.d.f. f(x,y). Let U=aX+b and V=cY+d, where a,b,c,d are fixed constants with a>0 andc>0. Show that thejoint p.d.f. of (U,V) isf(u,v)=f ()Thank you in advance.
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x = 0, 1, 2, . .. , zero elsewhere, where 0 0s I (a) Find the m. l.e., 6, of 0. X, is a complete sufficient statistic for 0. (b) Show that (c) Determine the unbiased minimum variance estimator of 0. 362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0),...
362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0), x = 0, 1, 2, . .. , zero elsewhere, where 0 0s I (a) Find the m. l.e., 6, of 0. X, is a complete sufficient statistic for 0. (b) Show that (c) Determine the unbiased minimum variance estimator of 0. 362 Suficient Statisties 7.76. Let X, X,.. ,x, be a random sample froma distribution with p.d.f. fix: 0) 0(1-0),...
6. Let Y be a random variable with p.d.f. ce4y for y 2 0 (a) Determine c. (b) What is the mean, variance, and squared coefficient of variation of Y where the squared coefficient of variation of Y is defined to Var(Y)/(E[Y])2? (c) Compute PríY <5) (d) Compute PríY >5 Y >1) (e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that PriY >
Let X and Y have a joint p.d.f. defined asf(x,y) = 1/2, if x+y<=2; x>=0; y>=00, elsewherea) Find the marginal p.d.f. of X and Y.b) Find P(X<=3/2; Y<=3/2)My solutions:a)Are the marginal p.d.f.s the following:fx(x) = integral(1/2, from 0 to (2-x)dy)fy(y) = integral(1/2, from 0 to (2-y)dx)
4. (25 pts, 25/6 pts each) Let X and Y be random variables of the continuous type having the joint p.d.f. f(x, y) = 8xy,0 £ x £ y £ 1. 1) Draw a graph that illustrates the domain of this p.d.f. 2) Calculate the marginal p.d.f.s of X and Y. 3) Compute 4) Compute 5) Write out the equation of the least squares regression line and draw it in a graph. 6) If your calculations are correct, in 3)...
7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhere, where 0 e< ao: (a) Find the m.l.e., 6. of 0. Is 6 unbiased? X and then compute E(0). Hint: First find the p.d.f. of Y = (b) Argue that Y is a complete sufficient statistic for 8. (c) Find the unbiased minimum variance estimator of 0. (d) Show that X/Y and Y are (e) What is the distribution of...
0, otherwise Let f(x,y)= 2. a. Sketch the region of integration b. Find k c. Find the marginal density of X d. Find the marginal density of Y e. Find P(Y > 0/X = 0.50) 0, otherwise Let f(x,y)= 2. a. Sketch the region of integration b. Find k c. Find the marginal density of X d. Find the marginal density of Y e. Find P(Y > 0/X = 0.50)