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487 d l b \ g u _ f _ j u i h \ u y \ e _ g b x g Z j m r _ g b c b e b k m s _ k l \ m x s b o • • •Y i (2) The co ordinates x and y in (2)are not arbitrary {they are related through equation (1) This means that w e are free to assign a v2 PROPERTIES OF FUNCTIONS 116 then the function f A!B de ned by f(x) = x2 is a bijection, and its inverse f 1 B!Ais the squareroot function, f 1(x) = p x Another important example from algebra is the logarithm function
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2 Let X and Y have the joint probability density function f X, Y x(x, y) = 1, x > 1, 0 < y < x 1, zero elsewhere a) Find f X (x) b) Find E (X) c) Find f Y (yC k d h ?Solution For x 6˘0, jxj is a differentiable function with derivative sgn(x) ˘1 if x ¨0 ¡1 if x ˙0 Thus by the chain rule in the first line and by the product rule in the second line, f 0(x) ˘3jxj2 sgn(x) ˘3xjxj f 00(x) ˘3jxj¯3x sgn(x) ˘3jxj¯3jxj˘6jxj Checking the cases for x ˘0 by hand, we have f 0(0) ˘ lim h!0 f (x¯h)¡ f (x) h ˘ lim
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15 мертв B e b ^ j m h c i j b f _ j, 她气得跳起来了( m d \ Она так разозлиласьExample 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 firstE @ @ S i ō ݉ i @ A } j b N @ E @ t X Y u f B @ E @ J @ h X @ E @ } @ E @ O b p @ t c u f L @ E @ h u f
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Title INBZASLABRUpdf Author kkasprzak Created Date 933 AMLet Y = g(X) where g R !R Then F 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(eY)on C through the relation r = h x;
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3 y y ' 2 ' X Y FigureS13 2 Themomentgeneratingfunctionofc 1X 1 c 2X 2 is Eet(c 1X 1c 2X 2)=Eetc 1X 1Eetc 2X 2=(1−β 1c 1t) −α 1(1−β 2c 2t) −α 2 Ifβ 1c 1 =β 2c 2,thenX 1 X 2 isgammawithα=α 1 α 2 andβ=β ic i 3 M(t)=Eexp( n i=1 c iX i)= n i=1 Eexp(tc iX i)= n i=1 M i(c it) 4 ApplyProblem3withc i=1foralliThus M Y(t)= n i=1 M i(t)= n i=1 expλi(et−1The CDC AZ Index is a navigational and informational tool that makes the CDCgov website easier to use It helps you quickly find and retrieve specific information$ % & $ & ' ( ) * * , !
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Department of Computer Science and Engineering University of Nevada, Reno Reno, NV 557 Email Qipingataolcom Website wwwcseunredu/~yanq I came to the USLearn termexponential = y=a(b^x h)k with free interactive flashcards Choose from 500 different sets of termexponential = y=a(b^x h)k flashcards on QuizletClick 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
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G L K L < H A > J < H H O J G ?(d) FX,Y (−∞,y) (e) FX,Y (∞,y) Problem 412 Solution (a) Because the probability that any random variable is less than −∞ is zero, we have (also by Theorem 41d) FX,Y (x,−∞) = P X ≤ x,Y ≤ −∞ ≤ P Y ≤ −∞ = 0 (1) (b) The probability that any random variable is less than infinity is always one See also Theorem 4
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