








(2) Suppose the random variables Yi and Yg have joint probability density function (n 2)-10 The marginal distributions are fi (y) = y/2 for 0 yIS 2 (zero otherwise) and fn (Y2)-2-2y2 for 0 Y2 1 (zero...
2. Suppose that Y and Y2 are continuous random variables with the joint probability density function (joint pdf) a) Find k so that this is a proper joint pdf. b) Find the joint cumulative distribution function (joint cdf), FV1,y2)-POİ уг). Be y, sure it is completely specified! c) Find P(, 0.5% 0.25). d) Find P (n 292). e) Find EDY/ . f) Find the marginal distributions fiv,) and f2(/2). g) Find EM] and E[y]. h) Find the covariance between Y1...
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
3. Let yi and ya have the joint density function otherwise, the same as in the previous problem. a) Show that yi and Y2 are dependent random variables. b) Note that when the joint density can be written as the product of a function of n and a function of 32 - which is the case here- the 2 random variables would be independent if the joint density is nonzero on a rectangular domain, according to a theorem we learned....
Let Yı, Y, have the joint density S 2, 0 < y2 <yi <1 f(y1, y2) = 0, elsewhere. Use the method of transformation to derive the joint density function for U1 = Y/Y2,U2 = Y2, and then derive the marginal density of U1.
5. If two random variables X and Y have the joint density k(52+2y2) for 0<<2 0 <y< 1 f(r, y) elsewhere (a) Find k (b) Find P(0<x< 1, 0<Y<0.5) (c) Find marginal density fi(a) and f2(y) (d) Are X and Y independent? (e) Find E(X) () Find P(X2 0.5). expression for fi(x|y); (g) an
Consider two random variables with joint density fY1,Y2(y1,y2)
=(2(1−y2) 0 ≤ y1 ≤ c,0 ≤ y2 ≤ c 0 otherwise
(a) Find a value for c. (4 marks) (b) Derive the density function
of Z = Y1Y2. (10 marks)
. Consider two random variables with joint density fyiy(91, y2) = 2(1 - y2) 0<n<C,0<42 <c o otherwise (a) Find a value for c. (4 marks) (b) Derive the density function of Z=Y Y. (10 marks)
The continuous random variables, X and Y , have the following joint probability density function: f(x,y) = 1/6(y2 + x3), −1 ≤ x ≤ 1, −2 ≤ y ≤ 1, and zero otherwise. (a) Find the marginal distributions of X and Y. (b) Find the marginal means and variances. (c) Find the correlation of X and Y. (d) Are the two variables independent? Justify.
Let Xi and X2 be two continuous random variables having the joint probability density f,2)10 0, elsewhere. a. the joint pdf o1% and Y2.9(Y1,Y2), b, the P06 > Yi), c. the marginal pdfs gn () and g2(2), d. the conditional pdf h(walvi), and e. the E(Yalki-y) and E(gYi = 1/2).
2. Suppose X and Y are continuous random variables with joint density function f(x, y) = 1x2 ye-xy for 1 < x < 2 and 0 < y < oo otherwise a. Calculate the (marginal) densities of X and Y. b. Calculate E[X] and E[Y]. c. Calculate Cov(X,Y).
et Yi and Y, be continuous random variables with the following joint probability density function 0, elsewhere. (a) Find E(Y1Y ) and E(YY-2) (b) Find the CDF and pdf of U mYo/Y. Your work should include a graph that supports your computatio Specify the domain where the pdf is positive.