Homework Answers
(2) Prove that the following are equivalent for x ER and A CR. (a) X E...
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Problem 3: Conditional Probabilities Let A and B be events. Show that P(An B | B)...
Problem 3: Conditional Probabilities Let A and B be events. Show that P(An B | B) = P(A | B), assuming P(B) > 0.
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z)...
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
5. Let S be a non-empty bounded subset of R. If a > 0, show that...
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n...
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint....
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X....
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X. Find the pdf of Y
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) =...
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
For s > 0 define the gamma function I (s) by T () = [co-dt. Show...
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
How to solve it? Let F =< -2, x, y2 >. Find S Ss curlF.nds, where...
How to solve it?
Let F =< -2, x, y2 >. Find S Ss curlF.nds, where S is the paraboloid z = x2 + y?, OSz54.
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Problem 3: Conditional Probabilities Let A and B be events. Show that P(An B | B) = P(A | B), assuming P(B) > 0.
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
Let s = {k=1CkXAz be a simple function, where {A1, A2, ... , An} are disjoint. Prove that for every p>0, |CK|PXAR
3. Suppose X ~ Beta(a, β) with the constants α, β > 0, Define Y- 1-X. Find the pdf of Y
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
How to solve it?
Let F =< -2, x, y2 >. Find S Ss curlF.nds, where S is the paraboloid z = x2 + y?, OSz54.
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