Problem 3 For μ>1> 0, let X₁ - Exp(1) and X₂ - Exp(u) be independent random variables. Show that the density of X₁ + X2 has the form f(x)= Auxe *1(0,00)(x), for some ye [A,μ].
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![Problem 3
For μ>1> 0, let X₁ - Exp(A) and X2 - Exp(u) be independent random
variables. Show that the density of X₁ + X2 has the form
f(x) = Auxe *1 [0,00) (x),
for some ye [A, μ].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4575c695-56bc-4a6a-843f-ec886ca258f2%2F12008cf2-ec46-4adc-991b-12af0fa4f7cd%2Fx7h0uf_processed.jpeg&w=3840&q=75)
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- Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y3. Let X be a continuous random variable. Let f(x) = c(x − 1)³ and Sx = [1,3]. Hint: (x - 1)³ = x³ + 3x − 3x² - 1 (a) What value of c will make f(x) a valid density? (b) What is P(X = 2)? (c) Find E(X). (d) What is P(1 < X < 2)?Let X and Y be two continuous random variables with joint probability density [3x function given by: f(x,y)= 0Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/201) The joint probabílity densíty functlon of the X and y random variables, cp fx,y (xiy)= cx,08. The density function for the random variable X is f(x). Find E(3X - 1). ) = √(= (2x + 1 (2x + 1) E(3X - 1) = f(x) = 1 < x < 2 0 elsewhere4. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 0Q1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=0Suppose X and Y are independent, Cauchy random variables with PDFS specified by 1/n fx(x) = 1+x2 , and fy(y) = 1+y2 1/n Find the joint PDF of fzw(z,w) where Z = x² + Y²,W = XYRecommended textbooks for youAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning