J1(1, 2) = log (₁) exp(-1₂) + sin(7₁) sec(1₂)f2(11, 12) = 12 - 4x1027, Consider o matrid of two Gaussian distributions as followsp(X₁₁ X₂) = 0.3 N ( [²2₂]₁ [ 02 1 ]+0.7 N ( [0] [005 001.01) What is the marginal distribution of X..b.) What is the marginal distribution of X₂.Ⓒ Compute ECX, ) and E(X₂)ol.) (ompute V(X₁) and V(X₂), the variances of Kianditz.5106. 7/ 87

0.3 and 0.7 are the mixing weights, representing the proportion of the population belonging to each…

Answered: J2 (21, 1₂) = 1₁ 1₂-4711²2 Consider o…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Suppose the rate of inflation in year t, It, is to be modelled by the AR(1) It = 0.0182 = 0.42(It-1 – 0.0182) + 0.042Zt where Z~N[0,1] The current (period 0) annual rate of inflation is 5.3% (1 = 0.053). b) Find the distribution of I₁. Provide the necessary calculation.
S If X is a Poisson variate such that P(X = 1) 3/10 and P(X = 2) = 1/5 Find P(X= 0) and P(X = 3). =
10 Consider the moment generating function given by: exp(7t + 5*(t^2) ). To which continuous distribution/density function does this belong? Note exp(x) = e^x. For the instructor, this was question 30. Normal distribution with mean =5 and variance =7 Normal distribution with mean =7 and variance =10 Normal distribution with mean =7 and variance =5 Normal distribution with mean =7 and variance =100
b) Explain how X can be related to the exponential distribution by using a moment generating function argument.
Q5) A random process is given by X(t) =P+ Q cos(@t+ 0) where P, Q are independent random variables. The mean and variance of distribution of random variable P are 0 and 25 respectively. The mean and variance of distribution of random variable Q are 0, 36 respectively. The angular frequency o is constant and 0 is independent of X(t). The PDF of 0 is given by f(0) = 1 2n 1/2 <0< 31/2. Prove that the random process X(t) is a wide sense stationary random process.
How do I find the expected value of x given its PDF and an unknown positive normalizing constant?
y 2 4 1 ГО.1 0.157 P(X, Y) = x3 0.2 0.3 %3D 5 Lo.1 0.15] a) Evaluate the marginal distribution of X and Y. b) Find P(Y/X) and P(X/Y). c) Find P(Y=2/X=3).
Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: 2g2 + xt = + ut 25 2g2
3. Let X; N iid U(0, 1) for i e {1, 2, ...,n}. (a) Compute the pdf of Y; = - log(X;), name the distribution that Y; follows and provide any parameter value or values needed. (b) Compute the pdf that W = [I", X, = X1 X2... X, follows. You may use the result given as part of the Example 2.6 video (even though it will only be proved in Chapter 3). (c) Compute E[W] in the two following ways: (i) through exploiting independence of the X, and using EUV] = E[U]E[V] for any independent random variables U and V. (ii) using the pdf of W computed in the previous part.
(c) What is the asymptotic distribution of √n(0-0)? 6.2.9. If X1, X2,..., Xn is a random sample from a distribution with pdf ={ f(x; 0) = 303 (x+0)4 0 0
Let x~ possion(alpha). Show that E[x(x-1)(x-2)....(x-k)]=ak+1
EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({0

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