Let X and Y be two random variables with joint pdf: f_{x,y}(x,y) =

(a) Find k such that f_{x,y}(x,y) is a joint pdf

(b) Find the marginal pdf fx(x) of X and cdf F_{x}(x) of X

(c) Find P(X + Y < 1)

(d) Let W = 5X - 3. Find the pdf f_{w}(w) of W

(e) Find E(5X - 3) and Var(5X - 3)

Community Answer

Solution: Let x&1 be 2 r.v.s with jo/nt por.f_(xy)(x,y)={[kxy,10 < x-y < 1],[0,10.w.]:}a) int_(0)^(1)int_(1)^(2)kxydydx=1{:[kint_(0)^(j)x[(y^(2))/(2)]_(0)^(1)+=1],[(k)/(2)[1-alpha][(x^(2))/(2)]_(d)^(1)=1],[(k)/(4)[1-d]=1]:}k=51b) plorginal pdf of xfx(x)=int_(0)^(1)4xydy{:[=4x","[(y^(2))/(2)]_(d)^(1)],[=2x[1-alpha]]:}{:[fx(x)=2x quad j0 < x < 1],[" cdf of "x],[Fx(x)=int_(0)^(x)2xda],[=2[(x^(2))/(2)]_(0)^(x)],[=x^(2)-0],[sqrt(Fx(x))=x^(2)]:}c){:[P(x+1 < 1)=int_(0)^(2)int_(0)^(1-x)4xydydx],[=4int_(0)^(1)x[(y^(2))/(2)]_(0)^(1-x)dx],[=2int_(0)^(d)x[(1-x)^(2)-d]dx],[=2int_(0)^(2)x(1+x^(2)-2x)dx]:}{:[P(x+14))=2int_(0)^(1)x+x^(3)-2x^(2)dx],[=2[(x^(2))/(2)+(x^(4))/(4)-(2x^(3))/(3)]_(0)^(1)],[=2((1)/(2)+(1)/(4)-(2)/(3))=2((3)/(4)-(2 ... See the full answer