Concept explainers
Let x and y be random variables of the continuous type having the joint
Draw a graph that illustrates the domain of this pdI.
(a) Find the marginal pdfs of X and Y.
(b) Compute
(c) Determine the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively?
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Probability And Statistical Inference (10th Edition)
- Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?arrow_forwardlet X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)arrow_forwarddW is normally distributed, dW has mean zero, dW has variance equal to dt. Parameter other than dw is assumed as constant. We have a representation of the geometric Brownian motion as dS/ S = µ dt + σ dW, prove µ dt + σ dW is normally distributed and find its mean and variance.arrow_forward
- Let x and y be joint continuous random variable with joint pdf f XY (x, y) = { cx+ 1, x, y≥ 0, x+y< 1 0, otherwise 1. Find the constant c. 2. Find the marginal PDF’S fX (x) and fY (y) 3. Find P(Y<2X^2 )arrow_forwardLet yt = φyt−1 + et with et ∼ WN(0,σ2) and |φ| < 1. Consider the over-differenced process wt = (1 − L)yt.(i) What is the model followed by wt? (ii) Is wt invertible? (iii) Obtain V [wt] and compare its magnitude with V [yt] and hence comment on the impact of over-differencing on the variance of a stationary process.arrow_forwardLet X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectivelyarrow_forward
- Let X and Y be jointly continuous random variables with joint PDF f X , Y ( x , y ) = { 6 x y , 0 ≤ x ≤ 1 , 0 ≤ y ≤ square root of x 0 , o t h e r w i s e . Then the marginal PDF f X ( x ) = { A , 0 ≤ x ≤ 1 , 0 , o t h e r w i s e . Find the function A.arrow_forwarda. What is the joint marginal PDF of X and Z?b. What is [P 0 <= Y <= 3/2 | X = 1/3 Z = 5/2]?arrow_forwardLet random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.arrow_forward
- f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.arrow_forwardLet X and Y be jointly continuous random variables with joint PDF f X , Y ( x , y ) = { 6 x y , 0 ≤ x ≤ 1 , 0 ≤ y ≤ square root of x 0 , o t h e r w i s e . Then the marginal PDF f Y ( y ) = { B , 0 ≤ y ≤ 1 , 0 , o t h e r w i s e . Find the function B. (a) 3 y (b) 3 y2 (c) 3 y ( 1 − y2) (d) 3 y ( 1 − y4 )arrow_forwardLet X be a continuous random variable with a pdf f(x) = { kx5 0≤x≤1, 0 elsewhere Determine the value of k.arrow_forward
- A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON