Page 129 Exercise 5: Suppose that the joint p.d.f. of two random variables X and Y is as follows: { c(x. 0 otherwise. ( 1 = c. = c

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1 Stat Solutions for Homework Set Page 9 Exercise : Suppose that the joint p.d.f. of two random variables X and Y is as follows: { cx fx, y + y for y x, < x < otherwise. Determine a the value of the constant c; b Pr X /; c PrY X + ; d PrY X. Solution: We must integrate over the support, y x. Thus But, c Thus c. fx, ydxdy yx + y x x dx c cx + ydydx c x x x + x dx c c x x c + c c x + ydydx c. Solutionb: Now the support has changed. Thus, along similar lines as in parta, P r X Solutionc: Here also support has changed. Thus / x x + ydydx x P r Y X + x+ x + ydydx 6. x x + x + x dx Solutiond: The probability the X, Y will lie on the curve y x is for every continuous joint distribution. Page 9 Exercise 6: Suppose that a point X, Y is chosen at random from the region S in the xy-plane containing all points x, y such that x, y, and y + x. a. Determine the joint p.d.f. of X and Y. b. Suppose that S is a subset of the region S having area α and determine P r[x, Y S ]. Solutiona: First, we find the area of the triangle that defines the support: /Base Height /. Thus since the area of S is, and the joint p.d.f. is to be constant over S, then the value of the normalizing constant must be /. Solutionb: The probability that X, Y will belong to any subset S is proportional to the area of that subset. Therefore, P r[x, Y S ] S dxdy area of S α. Page 9 Exercise 8: Suppose that X and Y are random variables such that X, Y must belong to the rectangle in the xy-plane containing all points x, y for which x and y. Suppose also that the joint c.d.f. of X and Y at every point x, y in this rectangle is specified as follows: F x, y 6 xyx + y.

2 a. Determine Pr X, Y. PrX, Y PrX, Y [PrX, Y PrX, Y ] [PrX, Y PrX, Y ] [F, F, ] [F, F, ] {[ ] [6 ]} b. Determine Pr X, Y. Pr X, Y since Pr X [F, F, ] [F, F, ] {[6 66] [6 ]} c. Determine the cumulative distribution function of Y. F Y y lim F X,Y x, y F X,Y, y x 6 xyx + y x 6 y + y y9 + y, if y < so that F Y y y9 + y if y. if y > d. Determine the joint p.d.f of X and Y. For < x < and < y < we exclude the boundary of the rectangular support set because these are sharp corner points and, hence, F x, Y is not differentiable at those points. f X,Y x, y F x, y [ ] x y x y 6 xyx + y 6 [ x + xy ] x 6 x + y. Then for x, y R \[, ] [, ] F x, y x y because F x, y or. At the boundaries we set fx, y arbitrarily. This doesn t matter since the boundaries represent sets of probability zero. Therefore the joint p.d.f of X and Y is e. Determine PrY X. fx, y 6 x + yi,, x, y. We need to find the probability of being in the region denoted by A for this joint density, namely: PrY X x x + ydydx x y + y x dx x + x dx [ x 6 ] [ x ] 79 9/8.7. 6

3 Page 9 Exercise : Let Y be the rate calls per hour at which calls arrive at a switchboard. Let X be the number of calls during a -hour period. A popular choice of joint pf/p.d.f for X, Y in this example would be one like: { y x f X,Y x, y e y if y > and x,,..., otherwise a. Verify that f is a joint pf/p.d.f. To verify that this is a valid probability function, we need to show that if we sum it over all values of x and integrate it over the region S y {y : y > }, we get the value. Doing this: y x f X,Y x, ydy e y dy e y y x dy b. Find PrX. x x x e y e y dy recognizing the power series for e y e y dy e y, as required. y PrX f X f X,Y, ydy e y dy! e y / /. e y dy Page Exercise : Suppose that X and Y have a discrete joint distribution for which the joint p.f. is defined as follows: { fx, y x + y for x,, and y,,,, otherwise a. Determine the marginal p.f. s of X and Y. The marginal p.f. for X is: PrX x f X x f X x f X,Y x, y y y x + y x + y [x + + x + + x + + x + ] y The marginal p.f. for Y is: PrY y f Y y b. Are X and Y independent? x + 6 x +, x,,. f Y y f X,Y x, y x x x + y x + y [ + y + + y + + y] x y + y +, y,,,. For discrete random variables X, Y to be independent, we must have: PrX x, Y y PrX x PrY y, for all x, y.

4 Let x, y. Computing: PrX 7/, PrY 6/, and PrX, Y /. But PrX, Y / 7/6/ PrX PrY, so X and Y are not independent. Page Exercise : Suppose that the joint p.d.f of X and Y is as follows: { fx, y x for y x and x otherwise a. Determine the marginal p.d.f s of X and Y. The marginal p.d.f for X is: f X x y f X x x f X,Y x, ydy x y x x x dy for x. Therefore, x x. Since y x, then x y y x y, y, then the marginal p.d.f for y Y is: f Y y x dx. That is f Y y x y y / + y / y y/, y. b. Are X and Y independent? Since the limits of integration of the joint p.d.f of X, Y, namely y x, cannot be stated separately for X and Y, X and Y are not independent. Putting it another way, since the support of X & Y is not the cross product set of the support of X x and the support of Y y, then X and Y are not independent. Page Exercise 6: a Since X and Y are independent, fx, y f xf y gxgy 9 6 x y I {r,s x, y }. b Since X and Y have a continuous joint distribution PrX Y. c Since X and Y are independent with the same probability distribution, PrX > Y PrY < X. Since PrX Y, it follows that. PrX > Y PrY < X, and PrX > Y.. d PrX + Y PrX Y and PrX + Y y fx, y, dxdy 8. Page Exercise : a fx, y is constant over the circle S. The area of S is π units, and so fx, y /π for x, y S and everywhere else. The possible values of X are in the interval [, ], and given some value of x [, ], x y x. Hence, f x x x dy π π x I [,] x.

5 By symmetry, f y f y X and Y have the same marginal distributions. b X and Y are not independent because fx, y f xf y. Question : Determine P Y X and P Y X. P Y X.9/..8 and P Y X./..67. Question : Determine PrX < Y and PrX < Y. PrX < Y.98 and PrX < Y.6.