Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX <.. Compute µ and σ as accurately as possible. Answer : Fortunately, symmetry helps out in this situation: we can determine immediately that µ. If PX > 9 PX <, then the problem would be trickier. After that, we can use Φ.8.. We conclude that µ/σ.8, or σ.48.. Suppose that X is a random variable with probability density function { e ax x < fx e bx x > where a and b are positive constants. a Show that a + b. b Compute Fx, the cumulative distribution function for X. c Compute E[X]. d Compute VarX. Answer : a Because fx dx, we see that b If y <, then If y >, then Fy c We have E[X] e ax dx + e ax dx + Fy y xfx dx ] ] e bx dx a e ax b e bx y e ax dx a e ax] y e ay /a. e bx dx a + b e bx ] y xe ax dx + xe bx dx a b. a + b b e by. d We have E[X ] ] ] a xe ax a e ax b xe bx b e bx b a b a b + a b a. x fx dx x e ax dx + x e bx dx a + b
] ] a x e ax a xe ax + a e ax b x e bx b xe bx b e bx a + b b + a b + a b ab + a b ab + a VarX E[X ] E[X] b ab + a b + ab a b + a.. The joint probability density function of the continuous random variables X and Y is given by x + xy < x <, < y < fx, y otherwise. a Verify that this is indeed a probability density function. b Compute the density function for X. c Compute P{X > Y}. d Compute P{Y > X < }. e Compute E[X]. f Compute E[Y]. Answer : a We have x + xy dy dx ] x y + xy dx x + x dx 4 ] x + x +. b If x < or x >, then f X x. For < x <, we have f X x fx, y dy x + xy dy ] x y + xy 4 x + x. c You need to be a bit careful about specifying the bounds of integration; drawing a graph is helpful. The region is < y < x <. We have x PX > Y x + xy dy dx ] x x y + xy dx 4 x + x dx 5 ] 4 4 x4 5 4 5. d We know that P{Y > X < } PY >, X < /PX <. Because we computed f X x, the denominator is straightforward: PX < f X x dx x + x dx x + x ] + 5 8 8.
The numerator comes from the denition: PY >, X < x + xy x + 5x Therefore, P{Y > X < } 9/448 9.85. 5/8 8 e We have f We have x x + xy E[X] dy dx ] x y + x y dx 4 ] x 4 + x 5. E[Y] y x + xy dy dx ] x y + xy dx ] x + x 8. dy dx dx x ] x y + xy dx 4 + 5x ] 9 448. x + x y dy dx x + x dx x y + xy dy dx x + 4x dx 4. Let fx, y /x if < y < x < and otherwise. Let fx, y be the joint density function for random variables X and Y. a Show that fx, y is a joint density function. b Compute the marginal density of X. c Compute the marginal density of Y. d Compute E[X]. e Compute E[Y]. Answer : a We have b We have fx, y dy dx f X x for < x <, and f X x otherwise. x x x dy dx /x dy dx.
c We have f Y y y ] /x dx log x log y y for < y <, and f Y y otherwise. Note that log y is positive for y between and. d We have e We have E[Y] E[X] yf Y y dy xf X x dx y log y dy x dx. y 4 y log y 5. Suppose that the joint density function for X and Y is given by { xe xy+ x >, y > fx, y otherwise. ] 4. a Verify that this is indeed a probability density function. b Find the conditional density function for X, given that Y y, and the conditional density function for Y, given that X x. c Find the density function for Z XY. Answer : a We have fx, y dy dx xe xy+ dy dx ] y e x e xy dx y xe x e xy dy dx e x dx. b The formul are f X Y x y fx, y/f Y y and f Y X y x fx, y/f X x, so really all we are being asked to compute here are the marginal density functions f X x and f Y y. We have f X x f Y X y x xe xy+ f Y y xe xy+ dy e x xe x e xy e x ] y xe x e xy dy e x e xy e x xe xy xe xy+ dx xe xy+ y + f X Y x y xe xy+ /y + y + xe xy+ ] x e xy+ y + x y y + c The usual approach works. First, compute the cumulative distribution function F Z a. Then dierentiate to get the probability density function f Z z. Because X and Y
are both positive, we know that F Z a if a < and f Z z if z <. For a >, we have F Z a PZ < a PXY < a PY < a/x a/x Therefore, f Z z F Z z e z. xe x e xy dy dx e x e a + e x dx e a. In class, we proved that if gx, y, then E[gX, Y] ] ya/x e x e xy dx y gx, yfx, y dx dy e x dx e a. Follow the argument in a previous homework to show that this formula is still correct for an arbitrary function gx, y. Answer : We have E[gX, Y] gx,y>t PgX, Y > t dt PgX, Y < t dt fx, y dx dy dt fx, y dx dy dt gx,y< t Set t u, u t, du dt in the second integral: fx, y dx dy dt x,y:gx,y>t Interchange the order of integration: gx,y dt fx, y dx dy x,y:gx,y> x,y:gx,y< x,y:gx,y> x,y:gx,y> gx,y du fx, y dx dy gx, yfx, y dx dy gx, yfx, y dx dy + x,y:gx,y<u x,y:gx,y< x,y:gx,y< fx, y dx dy du gx, yfx, y dx dy gx, yfx, y dx dy Adding in the x, y-region where gx, y contributes nothing to the sum: gx, yfx, y dx dy + x,y:gx,y> x,y:gx,y gx, yfx, y dx dy + x,y:gx,y< gx, yfx, y dx dy
gx, yfx, y dx dy x,y. Consider the experiment of rolling cubical dice, and recording the larger of the two numbers as a random variable X. What is the probability distribution for this random variable? What is E[X]? Answer : This can be done by counting all elements in the sample space. We get and E[X] 4.4. k PX k 5 4 5 9