**Exercise on Uniform Distribution and Histograms**Write a program to generate two independent and uniformly distributed random variables \( x_1 \) and \( x_2 \sim U(-1,1) \), each with 1000 realizations. (a) Use the histogram to verify the probability density function (pdf) of \( z = x_1 + x_2 \).(b) What shape does the histogram of \( w = \frac{1}{\sigma} \sum_{i=1}^{10} \frac{x_i}{2} \) look like? Can you explain why? Here, \( x_i \) is independent and identically distributed (iid) with a variance of \( \sigma^2 \).**Note:** Make sure to comment on the histogram's shape and its properties.

It is given that the random variable X follows Uniform distribution with parameters a (min)=-1 and b…

Answered: Write a program to generate two…

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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Use the result of Exercise 45 to show that, if n independent random variables Xi have normal distribu-tions with the means μi and the standard deviations σi, then Y = a1X1 + a2X2 +···+ anXn has a normal dis-tribution. What are the mean and the variance of this distribution?
dataset x = (x1, … , xn ) has sample variance of 0 when (n≥2 ). prove that all the data points are in fact equal.
A certain automobile manufacturer equips a particular model with either a turbo or a standard engine. Let X1 and X2 be fuel efficiencies for independently and randomly selected turbo and standard cars, respectively. Assume the two random variables are normally distributed with the following parameters: H = 22 and o,2 =1.44 (mean and variance of X1] Hi = 26 and = 2.25 [mean and variance of X2] Define a random variable Y as the difference between the first engine minus three times the second engine, X, - 3X, Find V(Y), the variance of the linear combination Y
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Vitamin B6 is one of the vitamins in a multiple vitamin pill manufactured by a phar- maceutical company. The pills are produced with a mean of 50 mg of vitamin B6 per pill. The company believes that there is a deterioration of 1 mg/month, so that after 4 months it expects that µ = 46. A consumer group suspects that < 46 after 4 months. Assume the amount of vitamin B6 per pill is normally distributed with unknown variance. : µ (i) Define a test for Ho = 46 against H₁ μ< 46 at a significance level a = 0.01 based : µ on a random sample of size n. (ii) If a sample of 20 pills yielded a mean of x = 45.88 with a (sample) standard deviation of s = 0.17, what is the conclusion based on your test defined in part (i)? Also, compute the associated p-value. (iii) Find the smallest significance level a under which we can reject Ho based on the data given in part (ii). Does it equal to the p-value found in part (ii)? Explain why.
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X is a discrete random variable such that the possible values are {a, a+1, a+2, ... b-1, b}. The random variable has a mean of 7 and a variance of 4. What is the Pr(X<=6|X>4)?
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The prevalence of mild myopia in adults over the age of 40 is 25% in the us supposed 4 adults over 40 are randomly chosen. Let Y denote the number of adult with myopia out of 4 with the following pmf of y Given X=10y-5 find the variance of X A) 75.10 B) 0.751 C) 70.10 D) 8.666
X and Y be standard normal random variables with correlation p. Compute the joint and marginal distributions of X+Y and X-Y. Are X+Y and X−Y independent?
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You are conducting a test of independence for the daim that there is an association between the row variable and the column variable. Red Blue Green Male 23 35 25 Female 23 13 60 What is the chi-square test-statistic and the corresponding p-value for this data? x² - [three decimal accuracy] p-value [three decimal accuracy]

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