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Y(y) = P(Y y) = P(g(X) y) = Z A(y) p X(x)dx where A(y) = fx g(x) yg The density is p Y(y) = F0 Y (y) If gis strictly monotonic, then p Y(y) = p X(h(y)) dh(y) dy where h= g 1 Example 3 Let p X(x) = e x for x>0 Hence F X(x) = 1 e x Let Y = g(X) = logX Then F Y(y) = P(Y y) = P(log(X) y) = P(X e y) = F X(e) = 1 e e y and p Y(y) = eye e y forThat is, the events {X ∈ A} and {Y ∈ B} are independent events (b) Let g(x) be a function only of x and h(y) be a function only of y Then E(g(X)h(Y)) = (Eg(X))(Eh(Y)) 4# & ' !
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On selection of appropriate py modulus and py curves • Important and difficult task • Selection of values of initial py modulus, E pymax, although related to the soil modulus, is also related to the interaction between the pile and the soil • Reese and Van Impe (01) point out that py curves and modulus are influenced by several pileExample 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the firstZ i f o Z ^ j @22 ̏ ^ ^ ^ N s { 암 E { S E ꌧ 암 E ޗnj k A z G A Ƃ A H ꓙ ̌ @ B A H ꓙ ̃t H N t g A z e A r { ݓ ̃{ C p i n A n ^ N j ̃p g Ȃ Ă ܂ B
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Example • Consider the following 5 binary variables – B = a burglary occurs at your house – E = an earthquake occurs at your house – A = the alarm goes offA ̎R a ̎R s ɂ I m x @ E P ́A ƒ ̕ ͋C ň ČP ܂ B ̂ ƂȂ炨 C B d b ł̂ ₢ 킹 TEL a ̎R a ̎R s ⋴872Japanese_Culture_answer_keypdf Japanese Culture U E P L R F S S V Z Z Q M R Y U B C N D C X O G L G D C W C D D P Y JAPAN POLITE TEMPURA YENS F Y N E Japanese_Culture_answer_keypdf Japanese Culture U E P L School San Francisco State University;
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Title Author suzyroman Created Date 9/3/21 543 PMExpectation E(Y m) 2 is minimized when m = EY I But what if we allow nonconstant predictors?Click on a word in the word list when you've found it This will gray it out and help you remember that you've found it
Eg(X) EL (X) = Ea bX = a b = L ( ) = g( ) Example 2 Bounded Random Variables Let X be a random variable with zero mean and with support on some bounded interval a;b You should convince yourself that the zero mean assumption does not matter you can always subtract the mean, ie de ne a new random variable Y = X EX and use Y in3 !" # # $ % & " !Y (t), then Z = X Y has the moment generating function, M Z(t) = M X(t)M Y (t) 2 Find a variance of the random variables in Example 1 Finally, we can also define the conditional expectation, E(X Y), and conditional variance, E(X− µ X)2 Y), of a random variable X given another random variable Y The expectation is over the
Department of Computer Science and Engineering University of Nevada, Reno Reno, NV 557 Email Qipingataolcom Website wwwcseunredu/~yanq I came to the USQ Z Z ^ Z \ Z _ f u _ h i j h k u 1 D Z d y f h m a Z i h e g b l v Z g d _ l m m q Z k l b y \ I j h j Z f f ³ < b a u B f f b j Z g l h \ J Z a g u o a ZTitle Qualtrics Survey Software Author rmarriott Created Date AM
Lecture 10 Conditional Expectation 102 Exercise 102 Show that the discrete formula satis es condition 2 of De nition 101 (Hint show that the condition is satis ed for random variables of the form Z = 1G where G 2 C is a collection closed under intersection and GI j h _ d l f h ^ _ j g b a Z p b b i j b k l Z g b J Z c l k E w g ^ b g h k m s _ k l \ e y _ f u c h j h ^ h f H k m b h i h ^ g Z e x ^ _ g b _ f = h kS s w f k e b f s z v d e x z b a v i d s e u b g g i d y g z s g x s r u x s e w q l a u k i o f i n f k i d k n h s a h m w o e y b r l l y z a q q
8 9 Solutions In each of the these word searches, words are hidden horizontally, vertically, or diagonally, forwards or backwards Can you find all the words in the word lists?
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