Problem 4.4. Determire the generating function for the following 4.4. Determir e the generating function for...
(10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y <1Y <2) (c) Find th e cumulative distribution function of Y, with domain R. (10 points) 4. The moment generating function of a random variable Y is , for t e R, where k is a constant. (a) Find the mean of Y. (b) Determine Pr(Y
Thus X has the probability mass function (for x1, x2, 3 E No with z +22 +3n) and moment generating function You do not need to show this.) i. Show that the moment-generating function of Y = (YĪ, defined as s given by 2 MARKS ii. Show that Yİ ~ Bi(n, 1 + 2) Hint: The probability mass function and moment-generating function of the binomial distribution can be found on the formula sheet.[3 MARKS] iii. One can also show that...
3. In this problem we consider only functions defined on the real numbers R A function f is close to a function g if r e Rs.t. Vy E R, A function f visits a function g when Vr E R, 3y E R s.t. For a given function f and n E N, let us denote by fn the following function: Below are three claims. Which ones are true and which ones are false? If a claim is true,...
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
Problem 7.1 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n]-1-0.2)" (b) x2[n] (0.3)" +2(-5) -0.2n Problem 7.2 (10 points) Express the unilateral z-transforms of the following functions as rational functions. Find also the ROC. You may use tables. (a) xl[n] = 3e-j02" (b) x2[n]- 5cos(5n) (c) x3[n] = e-0.gn sin(0.7n) Problem 7.3 (10 points) The signals given are sampled every 0.3 s, beginning att-0. Find...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
Problem 2.16 In this problem we explore some of the more useful theorems (stated without proof) involving Hermite polynomials. (a) The Rodrigues formula says that H (6) = (-1)” (1) . (2.87) Use it to derive H3 and H4. (b) The following recursion relation gives you Hn+1 in terms of the two preceding Hermite polynomials: Hn+1(E) = 2€ H, (E) – 2n Hn-1(5). (2.88) Use it, together with your answer in (a), to obtain Hg and H. (c) If you...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Model the following problem as a specified coefficient of an ordinary generating function. How many ways are there to choose eleven voters from a group of four voters from country A, six voters from country B and eight voters from country C if we wnat at least three country C voters in our selection? Assume that the voters of any country are indistinguishable or identical. a) Write down the equation that we want to solve. Explain all variables used. b)...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...