2.1 Solve the below problem (show all calculations).Let Y, and Y2 be random variables with joint density function0s y, < 1,0,2,0 s y2 <1, 0

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Answered: 2.1 Solve the below problem (show all…

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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Suppose that Y1, . . . , Yn is a random sample from a population whose density function is
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.)
2. Identify the probability density function, then find the mean and variance without integrating. b. f(x) =1/6 e^−x/6, [0,∞) c. f(x) =1 / 3√2π e^−(x−16)^2/18, (−∞,∞)
Suppose that the random variables X,Y, and Z have the joint probability density function f(x,y,z) = 8xyz for 0<x<1, 0<y<1, and 0<z<1. Determine P(X<0.7).
Suppose that the random variables X, Y , and Z have the joint probability density function f(x,y,z)=cxyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Find the E(x). Use the Scientific Method of Answering (Given, Required, Formula and Solution.)
Suppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independent
1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.
9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -
X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
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)= ?
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?

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