Consider a random experiment with S x y z t Let P x y t 08 and P y z 06 First show that it is a valid probability space Then use the Axioms of Probability and properties of Set Theory to compute the...

In this problem we will use the multi-variable chain rule to derive the (single variable) product rule We let f(t) and g(t) be differentiable functions and set h(t) = f(t)g(t) Let P(x y) = xy (a) Find the partial derivatives partP partx and partP party (b) Apply the multi-variable chain rule to P(x y) with x = f(t) and y = g(t) to find dP dt Your final answer should be expressed only in terms of the independent variable t (this is it shouldnrsquot contain x and y) (c) Show that h(t) = P(f(t) g(t)) and use part (b) to show dh dt (t) = df dt(t)g(t) + f(t) dg dt(t)

(1) Let f(u v)=u+v+u v2 and u=u(x y)=ex+y v=v(x y)=x y Use the Chain Rule to compute f x and f y (2) Repeat problem 1 but this time first finding the formula for the composite function f(u(x y) v(x y)) then computing f x and f y directly Check that these partial derivatives are equal to the ones found in problem 1 (3) Let z=ln ft(x2+y ) x=r et and y=t er Use the Chain Rule to compute z r and z t at the point where (r t)=(12) (4) Suppose a moth is flying in a circle about a candle flame so that its position at time t is given by x= t y= t Suppose that the air temperature is given by T(x y)=x2 ey-x y3 Use the Chain Rule to find a formula for d Td t the rate of change of the temperature the moth feels (5) Compute z x using the Chain Rule trick described in this section if (x y z)+x+2 y+3 z=0 (6) (Bonus) A rectangular box has its edges changing length as time passes At a particular instant the sides have lengths a=150 feet b=80 feet and c=50 feet At that instant a is increasing at 100 feet/sec b is decreasing 20 feet/sec and c is increasing at 5 feet/sec Determine if the volume of the box is increasing decreasing or not changing at all at that instant

Let C be the curve of intersection of the spheres x2+y2+z2=27 and (x-2)2+(y-2)2+z2=27 Find the parametric equations of the tangent line to C at P=(115) It is known that if the intersection of two surfaces F(x y z)=0 and G(x y z)=0 is a curve C and P is a point on C then the vector v= FP GP is a direction vector for the tangent line to C at P (Use symbolic notation and fractions where needed Enter your answers as functions of parameter t in a form r(t)= x(t) y(t) z(t) = r 0+ v t where r 0 is the corresponding coordinate of point P)

This exercise develops a useful identity for covariance similar in spirit to Adams law for expectation and Eves law for variance First define conditional covariance in a manner analogous to how we defined conditional variance Cov(X Y Z)=E((X-E(X Z))(Y-E(Y Z)) Z) (a) Show that Cov(X Y Z)=E(X Y Z)-E(X Z) E(Y Z) This should be true since it is the conditional version of the fact that Cov(X Y)=E(X Y)-E(X) E(Y) and conditional probabilities are probabilities but for this problem you should prove it directly using properties of expectation and conditional expectation (b) ECCE or the law of total covariance says that Cov(X Y)=E( Cov(X Y Z))+ Cov(E(X Z) E(Y Z)) That is the covariance of X and Y is the expected value of their conditional covariance plus the covariance of their conditional expectations where all these conditional quantities are conditional on Z Prove this identity Hint We can assume without loss of generality that E(X)=E(Y)=0 since adding a constant to an rv has no effect on its covariance with any rv Then expand out the covariances on the right-hand side of the identity and apply Adams law Solve and show steps