These questions are in most cases taken from Newbold et al. For numericalquestions, obtain an answer both with a computer program such as Matlab, R, orPython and with tables if you are able to do so, but solve with tables in any event.1. A random variable has the density function: fx(x) = 0, if x < 3; x − 3, if 3 ≤x ≤ 4; 5 – x, if 4 < x < 5; 0, x ≥ 5.Using the expression for the expectation (mean) of a continuous random vari-able, i.e.E(X) =xfx(x)dx,treating the two intervals 3 ≤ x ≤ 4 and 4 ≤ x ≤ 5 separately, show that the meanof this random variable is 4.2. (a) Using the expression for variance in either the discrete or continuous case,show that for a random variable X and fixed parameters a and b, var(a + bx) =b²var (X).(b) Using the expression K(X) = E(X−µ) as a kurtosis measure, show that fora random variable X and fixed parameters a and b, K(a + bX) = K(X). Interpretthis with respect to the standardization by ¹.3. Let X₁ and X₂ be a pair of random variables. Show that the covariance betweenthe random variables (X₁ + X₂) and (X₁ – X₂) is 0 if and only if X₁ and X₂ havethe same variance.

As per the Bartleby guildlines we have to solve first question and rest can be reposted... Given…

Answered: These questions are in most cases taken…

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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If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.
The proportion of people who respond to a certain mail-order solicitation is a random variable X having the following density function. f(x) = 2(x+1)/3, 0<x<1, 0, elsewhere Find σ2g(X) for the function g(X)=5X2+4. σ2g(X)= ?
For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
Given the probability mass function of a random variable X p(x) = c/(x+1) if x = 0,1 and p(x) = 0 in other mass points. What is the value of c ?
Suppose a random variable X has a probability density function as shown in img1.jpg c. Find P(X <= 0.4 | X <= 0.8) Note: I already solve part A and the value of k is 6.
The random variables X and Y have the following joint probability density function:f(x,y)={e−x−y , 0<x<∞; 0, elsewhere. What is Cov(X,Y)(X,Y)?
Here is another infectious disease model. Once a person becomes infected, the time X, in days, until the person becomes infectious (can pass on the disease) can be modeled as a Weibull random variable with density function f(x,α,β) = (α/βα)xα−1e−(x/β)α for 0 ≤ x ≤ ∞ and 0 otherwise with α = 3.7 and β = 7.1α is the shape parameter and β is the scale parameter. Hint: Solve this with the built-in R functions for the Weibull distribution (dweibull(),pweibull(), qweibull()) not f as defined above. Otherwise you may get intermediate values too large to use. For a) and b) the text (and notes) give formulas for the answers. You can calculate from these formulas. Note that these formulas use the gamma function. Γ(α) is the gamma function. In R, there is a built-in function gamma() which calculates this.d) What is the probability that X is larger than its expected value?e) What is the probability that X is > 2?
Here is another infectious disease model. Once a person becomes infected, the time X, in days, until the person becomes infectious (can pass on the disease) can be modeled as a Weibull random variable with density function f(x,α,β) = (α/βα)xα−1e−(x/β)α for 0 ≤ x ≤ ∞ and 0 otherwise with α = 3.7 and β = 7.1α is the shape parameter and β is the scale parameter. Hint: Solve this with the built-in R functions for the Weibull distribution (dweibull(),pweibull(), qweibull()) not f as defined above. Otherwise you may get intermediate values too large to use. For a) and b) the text (and notes) give formulas for the answers. You can calculate from these formulas. Note that these formulas use the gamma function. Γ(α) is the gamma function. In R, there is a built-in function gamma() which calculates this.f) What is the probability that X is > 4?g) What is the probability that X > 4 given that X > 2?h) Calculate the 70th percentile of X
Suppose that the unknown X is ≥ 2 and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?
Assume that the range of X is [4.9, 5.11 mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?
The PDF of a continuous random variable X is as follows: f(X)= c(4x2 - 2x2) 0<* x <* 2 (*less or equal to) a. For this to be a proper density function, what must be the value of c ?
The error, X, in measuring the length of a rivet (in micrometers) has the following probability density functionf(x)= 20·x·(3−x)3 243 for 0 ≤ x ≤ 3 and 0 otherwise.a) What is the probability that X > 1? b) What is the probability that X < 1? c) What is the probability that 2.1 < X < 2.7? d) What is the expected value of X (E(X))?

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