Suppose that X1 and X2 have a continuous joint

Suppose that X1 and X2 are i.i.d. random variables and that each of them has the uniform distribution on the interval [0, 1]. Find the p.d.f. of Y = X1 + X2

For the conditions of Exercise 1, find the p.d.f. of the average (X1 + X2)/2.

Suppose that three random variables X1, X2, and X3 have a continuous joint distribution for which the joint p.d.f. is as follows: f (x1, x2, x3) =  8x1x2x3 for 0 < xi < 1 (i = 1, 2, 3), 0 otherwise. Suppose also that Y1 = X1, Y2 = X1X2, and Y3 = X1X2X3. Find the joint p.d.f. of Y1, Y2, and Y3

Suppose that X1 and X2 have a continuous joint distribution for which the joint p.d.f. is as follows: f (x1, x2) =  x1 + x2 for 0 < x1 < 1 and 0 < x2 < 1, 0 otherwise. Find the p.d.f. of Y = X1X2.

Suppose that the joint p.d.f. of X1 and X2 is as given in Exercise 4. Find the p.d.f. of Z = X1/X2.

Let X and Y be random variables for which the joint p.d.f. is as follows: f (x, y) =  2(x + y) for 0 x y 1, 0 otherwise. Find the p.d.f. of Z = X + Y .

Suppose that X1 and X2 are i.i.d. random variables and that the p.d.f. of each of them is as follows: f (x) =  ex for x > 0, 0 otherwise. Find the p.d.f. of Y = X1 X2.

Suppose that X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1] and that Yn = max {X1,...,Xn}. Find the smallest value of n such that Pr{Yn 0.99} 0.95

Suppose that the n variables X1,...,Xn form a random sample from the uniform distribution on the interval [0, 1] and that the random variables Y1 and Yn are defined as in Eq. (3.9.8). Determine the value of Pr(Y1 0.1 and Yn 0.8).

For the conditions of Exercise 9, determine the value of Pr(Y1 0.1 and Yn 0.8).

For the conditions of Exercise 9, determine the probability that the interval from Y1 to Yn will not contain the point 1/3.

Let W denote the range of a random sample of n observations from the uniform distribution on the interval [0, 1]. Determine the value of Pr(W > 0.9).

Determine the p.d.f. of the range of a random sample of n observations from the uniform distribution on the interval [3, 5].

Suppose that X1,...,Xn form a random sample of n observations from the uniform distribution on the interval [0, 1], and let Y denote the second largest of the observations. Determine the p.d.f. of Y. Hint: First determine the c.d.f. G of Y by noting that G(y) = Pr(Y y) = Pr(At least n 1 observations y).

Show that if X1, X2,...,Xn are independent random variables and if Y1 = r1(X1), Y2 = r2(X2), . . . , Yn = rn(Xn), then Y1, Y2,...,Yn are also independent random variables.

Suppose that X1, X2,...,X5 are five random variables for which the joint p.d.f. can be factored in the following form for all points (x1, x2,...,x5) R5: f (x1, x2,...,x5) = g(x1, x2)h(x3, x4, x5), where g and h are certain nonnegative functions. Show that if Y1 = r1 (X1, X2) and Y2 = r2 (X3, X4, X5), then the random variables Y1 and Y2 are independent

In Example 3.9.10, use the Jacobian method (3.9.13) to verify that Y and Z are independent and that Eq. (3.9.18) is the marginal p.d.f. of Z.

Let the conditional p.d.f. of X given Y be g1(x|y) = 3x2/y3 for 0

Let X1 and X2 be as in Exercise 7. Find the p.d.f. of Y = X1 + X2.

If a2 = 0 in Theorem 3.9.4, show that Eq. (3.9.2) becomes the same as Eq. (3.8.1) with a = a1 and f = f1.

In Examples 3.9.9 and 3.9.11, find the marginal p.d.f. of Z1 = X1/X2 by first transforming to Z1 and Z2 = X1 and then integrating z2 out of the joint p.d.f.