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F x x g x hx x x y o o Note In part (a), hx can also be written as h x x 2 (b) 2 Left 2 f x x g x h xx x 2 y o o (c) No, parts (a) and (b) do not yield the same function, since z xx22 Both graphs are shown below to emphasize the difference in the final results (but we can see that the above functions are different without graphing the0, if x ≤ a, 51, if x ≥ b;

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ƒxƒŠ[ƒVƒ‡[ƒg ƒwƒA ƒXƒ^ƒCƒ‹ ƒƒ"ƒY 40 'ã ƒ~ƒfƒBƒAƒ€-(b) Sketch the cone and make a rough sketch of C on the cone solution x = t cost, y = t sin t and z = t, hence x2 y2 = t2 cos2 t t2 sin2 t = t2 cos2 t sin2 t = t2 = z2 x2 y2 = z2 is the equation of a circular cone, hence the curve lies on a circular cone As the height z = t increases linearly with time, the x and y coordinates traceU0= 2xu c for all real x For which value(s) of the real constant cis this set a linear subspace of C(R)?

Assignment Bioprocess

 AVL Tree Set 1 (Insertion) AVL tree is a selfbalancing Binary Search Tree (BST) where the difference between heights of left and right subtrees cannot be more than one for all nodes The above tree is AVL because differences between heights of left and right subtrees for every node is less than or equal to 1If x ∈ (a,b) 0, otherwise, cdf F(x) = x−a b−a, if x ∈ (a,b);∫∫ () √ Example 4 Let ≤ ≤ denote the order statistics of a random sample of size 3 from a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere

B) the marginal densities f X(x) and f Y(y) c) Are Xand Y independent?Let us start with a function, in this case it is f(x) = x 2, but it could be anything f(x) = x 2 Here are some simple things we can do to move or scale it on the graph We can move it up or down by adding a constant to the yvalue g(x) = x 2 C Note to move the line down, we use a negative value for C C > 0 moves it up;D) Find EXand EY SOLUTION The region is a square rotated by 45 degrees with corners at ( 1;0) and (0;

< 7 0 3?EC02 Spring 06 HW3 Solutions 3 Problem 224 • The random variable X has PMF PX (x) = ˆ c/x x = 2,4,8, 0 otherwise (a) What is the value of the constant c? The composition of f and g is the function g ∘ f A → C defined by (g ∘ f)(x) = g(f(x)) for all x ∈ A We often refer to the function g ∘ f as a composite function It is helpful to think of composite function g ∘ f as " f followed by g " We then refer

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C(x,v) = cost for direct link from x to v Node x maintains costs of direct links c(x,v) D x(y) = estimate of least cost from x to y Node x maintains distance vector D x = D x(y) y єN Node x maintains its neighbors'distance vectors For each neighbor v, x maintains D v = D v(y) y єN Each node v periodically sends D v to its neighborsGraph y=x Use the slopeintercept form to find the slope and yintercept Tap for more steps The slopeintercept form is , where is the slope and is the yintercept Find the values of and using the form The slope of the line is the value of , and the yintercept is the value of Slope(E611) so v x is independent of the distance from the inlet, and the velocity proflle will appear the same for all values of x Since @v x=@z= 0 (assumption 3), it follows that v x= v x(y) is a function of yonly Axis of symmetry L Exit Velocity profile Inlet Wall Wall

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Properties of variance 30Cross Product Definition If a = and b = , then the cross product of a and b is the vector, a x b =F ma F F N kg m s net y y net y g = ⋅ → = ⋅ = = − = ⋅ − ⋅ − − − 13 In the figure below, mblock =85kg and θ=30º Find (a) Tension in the cord (b) Normal force acting on the block (c) If the cord is cut, find the magnitude of the block's acceleration T N Fg y

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In this math video lesson I solve the equation g=xcy , for x This is a useful skill for students who are in Algebra and will help them to better understaAdded by ihsankhairir in Mathematics To obtain the composite function fg(x) from known functions f(x) and g(x) Use the hatch symbol # as the variable when inputting> 5 / 2;

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(eg P(Y = yjX= x)) Independence for rv's Xand Y This is a good time to refresh your memory on doubleintegration We will be using this skill in the upcom y 15 16 fY(y) 060 040 Because the the probability mass functions for X and Y appear in the margins of the table (ie column and row totals), they are often reDVDs ,x1 CDs ,x2 m p1 = m p2 = 40 10 15 Given that p 1 = 40, p 2 = , and m = 800, we can rewrite these two equations as (1) 40x 1 x 2 = 800 (2) 403x 2 x 1 = =)x 2 = 2 3 x 1 (d) To nd Alicia's optimal bundle, we just use the two equations above to solve for ourMath 461 Introduction to Probability AJ Hildebrand Variance, covariance, correlation, momentgenerating functions In the Ross text, this is covered in Sections 74 and 77

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Replacing x and y with x/w and y/w yields A(x/w) B(y/w) C = 0 Multiplying by w changes it to Ax By Cw = 0 Let the given equation be a second degree polynomial Ax 2 2Bxy Cy 2 2Dx 2Ey F = 0 After replacing x and y with x/w and y/w and multiplying the result with w 2, we have(121) y ay by g x where a and b are constants, and g x is a differentiable function of x In chapter 64, we saw that a first order equation has a oneparameter family of solutions, and that the specification of an initial condition y x0 y0 uniquely determines a solution In the case of second order equations, the basic theorem isC < 0 moves it down

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4 8 Let X be a discrete random variable taking on the two values ±10 with equal probability Let Y be a uniform random variable on the interval (1,1) If Z = X Y, and X and Y are independent, find the probability density function for the random variable Z 9 Consider a Gaussian random process X(t) with autocorrelation function a Find the total average power, E(X2)C1 = RL 0 f(x)dx¡ 7 2 L2 ¡ 5 24 L4 L;Search the world's information, including webpages, images, videos and more Google has many special features to help you find exactly what you're looking for

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³ (b) Please refer to the textbook /notes for the order statistics pdf and joint pdf formula We have ;Partial derivatives of composite functions of the forms z = F (g(x,y)) can be found directly with the Chain Rule for one variable, as is illustrated in the following three examples Example 1 Find the xand yderivatives of z = (x2y3 sinx)10And then ` = ¡ 1 6 x3 µ 7 1 2 L2 ¶ x RL 0 f(x)dx¡ 7 2 L2 ¡ 5 24 L4 L 152 For conduction of thermal energy, the heat °ux vector is ` = ¡K0ru If in addition the molecules move at an average velocity V, a process called convection, then ` = ¡K0ruc‰uV Derive the corresponding equation for

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574 Evaluate a triple integral using a change of variables Recall from Substitution Rule the method of integration by substitution When evaluating an integral such as ∫3 2x(x2 − 4)5dx, we substitute u = g(x) = x2 − 4 Then du = 2xdx or xdx = 1 2du and the limits change to u = g(2) = 22 − 4 = 0 and u = g(3) = 9 − 4 = 5V 8 = 0 V = 800 kip 6 ft 6 x 10 ft M = {x2 300x 216} kip # ft ©M NA = 0;(B) c f 2V Since f(x) > 0, we have that (c f)(x) = f(x)c is de ned for all x 2R, and furthermore, this is clearly positive Therefore, c f is in V (1) f g = g f To check that two functions are equal, we need to check that they are the same at all x 2R Thus, we need to

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F(x) = ax b, g(x) = cx dMicrosoft Teams LoadingThen VarY = (1)2VarX = 1 But XY = 0, always, so VarXY = 0 Ex 2 As another example, is VarXX = 2VarX?

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M 216 2xa x 2 b 300x = 0 V = {300 2x} kip c©F y = 0;1) The area of this square region is 2, so the density is f(x;y) = (1 2 0Free shipping on millions of items Get the best of Shopping and Entertainment with Prime Enjoy low prices and great deals on the largest selection of everyday essentials and other products, including fashion, home, beauty, electronics, Alexa Devices, sporting goods, toys, automotive, pets, baby, books, video games, musical instruments, office supplies, and more

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Definition The mutual information between two continuous random variables X,Y with joint pdf f(x,y) is given by I(X;Y) = ZZ f(x,y)log f(x,y) f(x)f(y) dxdy (26) For two variables it is possible to represent the different entropic quantities with an analogy to set theory In Figure 4 we see the different quantities, and how the mutualB) Let C2(R) be the linear space of all functions from R to R that have two continuous derivatives and let S f be the set of solutions u(x) 2C2(R) of the di erential equation u00 u= f(x) for all real x For which polynomials f(x) is the set SF(x,y,y0,y00)=0 This chapter is concerned with special yet very important second order equations, namely linear equations Recall that a first order linear differential equation is an equation which can be written in the form y0 p(x)y= q(x) where p and q are continuous functions on some interval I A second order, linear

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3 If z = f(x) for some function f(), then –z = jf0(x)j–x We will justify rule 1 later The justification is easy as soon as we decide on a mathematical definition of –x, etc Rule 2 follows from rule 11 uniform distribution on (a,b) With a and b constants, X has density function f(x) = ˆ 1 b−a;Given a vector v ∈ R2, let (x,y) be its standard coordinates, ie, coordinates with respect to the standard basis e1 = (1,0), e2 = (0,1), and let (x′,y′) be its coordinates with respect to the basis u1 = (3,1), u2 = (2,1) Problem Find a relation between (x,y) and (x′,y′) By definition, v = xe1 ye2 = x′u1 y′u2 In standard

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6 ft 4 ft 300 2x V = 0 0 x 6 6 ft 6–18 Draw the shear and moment diagrams for the beam, and determine the shear and moment throughout the beamIn general VarXY ≠ VarX VarY Ex 1 Let X = ±1 based on 1 coin flip As shown above, EX = 0, VarX = 1 Let Y = X;EC02 Spring 06 HW5 Solutions 3 Problem 321 • The random variable X has probability density function fX (x) = ˆ cx 0 ≤ x ≤ 2, 0 otherwise

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Assignment Bioprocess

Algebra Solve for x Calculator Step 1 Enter the Equation you want to solve into the editor The equation calculator allows you to take a simple or complex equation and solve by best method possible Step 2 Click the blue arrow to submit and see the result!The function f(x) is a probability density function for the continuous random variable X, de ned over the set of real numbers R, if 1 f(x) 0, for all x 2 R 2 R1 1 f(x)dx = 1 3 P(a < X < b) = Rb a f(x)dx Ex 9 on p 73 The proportion of people who respond to a certain mailorder solicitation is a continuous random variable X that hasP o l i c y a n d d o c t r i n e i n C h a i r m a n , J o i n t Chiefs of Staff Instruction D and J o i n t P u b l i c a t i o n 3 – 1 3 3 I n J o i n t a n d Army operations, operations security is an i n f o r m a t i o n r e l a t e d c a p a b i l i t y i n t e g r a t e d by Information Operations as prescribed

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8 Let W have the density function given by f W(w) = 2w for 0 < w < 1 and f W(w) = 0 for other values of w Set Y = eW (a) Determine the distribution function and quantiles of W (b) Determine the distribution function, density function, and quantiles of Y (c) Determine the mean and variance of Y directly from its density functionNote that (f B g)(x) ≠ (g B f)(x) This means that, unlike multiplication or addition, composition of functions is not a commutative operation The following example will demonstrate how to evaluate a composition for a given value Example 6 Find (f B g)(3) and (g B f)(3) if f ( x ) = x 2 and g ( x ) = 4 – x2 Solution Step 1 Find (fPage 2 c) d) wx x′y BD B′D′ A′B 312) Simplify the following Boolean functions in products of sums a) F(w, x, y, z) = ∑ (0, 2, 5, 6, 7, 8, 10)

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Here we will look at solving a special class of Differential Equations called First Order Linear Differential Equations First Order They are "First Order" when there is only dy dx, not d 2 y dx 2 or d 3 y dx 3 etc Linear A first order differential equation is linear when it can be made to look like this dy dx P(x)y = Q(x) Where P(x) and Q(x) are functions of x To solve it there is aEx) X and Y have joint PDF f(x,y) = c x y2 if 0 < x < y < 1 = 0 elsewhere • Find c First, draw the region where f > 0 (not 0 y x 1 1 cxy dxdy y ³ ³ 0 2 1 0 1 cxy dxdy y x ³ ³ 0 2 1 cxy dydx x 2³ ³ 1 1 0An online gof fog calculator to find the (fog) (x) and (gof) (x) for the given functions In this online fog x and gof x calculator enter the f (x) and g (x) and submit to know the fog gof function Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator

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@v x @x @v y @y @v z @z =0 (552) But since v y= v z=0 @v x @x =0;M 8(10 x) 40 = 0 c©F y = 0;0 3 0 & 1 2 * 3 * * 4 5 6 7 " 2 8

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