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Homework #1 1. Suppose that a variable Z(t) x(t)/y(t). Prove that for Zlt) to be constant,...
(a) Suppose that X, Y and Z are random variables whose joint distribution is continuous with density fxyz. Write down appropriate definitions of of (i) fxyz, density of the joint distribution of X and Y given Z, and (ii) fxyz, density of the distribution of X given both Y and Z. Assuming the expectations exist, prove the tower property: E[E[X|Y, 2]|2] = E[X|2], by expressing both sides using the densities you have defined. Suppose that X and Y are independent...
a) Prove that if Y is a random variable with all of its cumulants of order greater than two equal to zero, i.e., 0 = K_3 = K_4 = ......... then Y has a normal distribution. b) Suppose that Z has cumulant generating function k_Z(t) with E(Z) = 0 and Var(Z) = 1. Let Y = σZ + μ. Find the cumulant generating function of Y , k_Y (t) in terms of k_Z(.). Use this to prove that all cumulants...
Suppose that a rv Y has mgf m(t)- (a) 1-bt) Differentiate this mgf twice and thereby obtain the mean and variance of Y. [5 marksj] (b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t), ie·r(t) = login(1). Find r'() and r"(t), and express r'(0) and r"(0) in terms of EW and VarW. [5 marks] Use the result in (b) to find the mean (d) Find the mean and variance of the...
take the derivative of (x+y)^3 = 3x - 4y + 5 with respect to the variable t. do not rearrange and move terms to the other side of the equation.
Suppose s(t) is the arc-length parametrization of a space-turd flying through space. What is the arc-length of the space-turd's path between time t = 1 and t = 70 ? Question 10 1 pts Suppose f(x, y, z) = xy cos z.Compute the partial derivative of f with respect to the variable y at the point (4,2, 7).
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
In the previous homework, the Fourier Transform of x(t)- t[u(t)-u(t-1) was found to be x(t) 2 0 -1 -2 -3 5 4 3-2 0 2 3 4 5 a) b) Using known Fourier transforms for the terms of y(t), find Y(j). (Hint: you will have to apply some c) Apply differential properties to X(ju) to verify your answer for part b Differentiate x(t), y(t) = dx/dt. Note, the derivative should have a step function term. Include a sketch of y(t)...
The equation W = F(x, y, z) =0 defies the variable z implicitly as a function zz flxy). Draw a branch diagram for differentiating w with respect to x, then prove dz dx Ez
X Y Z iid Suppose for random variable X, P(X > a) - exp( random variable Y, P(Y > y) exp(-0y) for y > 0, and for random variable , P(Z > z)--exp(-фа) for z > 0. (a) Obtain the moment generating functions of X, Y and Z. (b) Evaluate E(X2IX > 1) and show it is equal to a quadratic function of λ. (c) Calculate P(X > Y Z) if λ-1, θ--2 and φ--3. -λα) for x > 0,...
Suppose that (X, |..|/x) and (Y, ||:y) are Banach spaces, and T : XxY – C is bilinear and separately continuous, i.e. T is continuous in z if y is fixed, and continuous in y if x is fixed. Prove that T' is jointly continuous, i.e. if 27 + 1* and if yn →y", then Tiera, Yn) + T(3*, Y").