2. Problem. Let X be a random variable such that E(X)= 0. Assuming that thevariance of X exists, show thatHint: Note thatP(X ≥a) ≤var (X)var (X) + a²for all a > 0.P(X≥a) ≤ P ((X+t)² ≥ (a+t)²)Apply the Markov inequality and optimize on t.for all t20.

Given that X is a random variable such that E(X)=0 and its variance, var(X) exists. We have to show…

Answered: 2. Problem. Let X be a random variable…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X1, X2, X3, and X4 be independent Poisson random variables, with parameters A₁ = 3A, A2 = A, A3 = 9A, and A4 = 4A, respectively. (a) Find and simplify the joint pmf f(x1, x2, x3, x4; A1, A2, A3, A4). (b) Find L(A) and (A), the likelihood and log-likelihood functions.
2. Let X ~ N(0, o²) and Y ~ (0, v²) be independent random variables and Z = X + Y. (a) Find the LMMSE estimator of X given Z. (b) Find the conditional distribution fz|x(z, x). (c) Using the previous part, find the conditional distribution fx|z(x, z). (d) Find the MMSE estimator of X given Z and show that it is the same as the LMMSE estimator in Part (a).
f (x) = 4xc for 0 ≤ x ≤ 1 ; Find the constant c so that f (x) is apdf of some random variable X, and then find the cdf, F (x) = P(X ≤ x). Sketch graphsof the pdf f (x) and the cdf F (x), and find the mean μ and variance σ2.
What holds for PY |X(y|x) and PX|Y (x|y) if the random variables X and Y are independent?3. It holds true thatEX[(x − 2)2] = EX[x2] − a · EX[x] + b.What are a and b?4. Prove that the maximum likelihood decision is maximum a posteriori, too, if PS(s)is constant.
Let X and Y be independent and N(0, 1) distributed random variables. Let U = X and V = X/Y. Show that the random variable V is Cauchy distributed and find E(V ).
Let x₁ = 6, x2 = 4, x3 = 5, x4 = 15 be a random sample from a population with probability density function a 10a (x + 10)α+1 Compute â, the maximum likelihood estimate of a. f(x) = You do not have to verify that your estimate is a maximum. 2 x > 0.

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2. Problem. Let X be a random variable such that E(X)= 0. Assuming that the variance of X exists, show that P(X ≥a) ≤ Hint: Note that var(X) var(X) + a² for all a > 0. P(X≥a) ≤ P((X+t)² (a+t)²) for all t≥ 0. Apply the Markov inequality and optimize on t.