Suppose that a point X is selected at random from the interval (0,1). After the value X = x has been selected, a point Y is then chosen at random from the interval (0,x^2). a) Indicate the region R on the xy-plane of possible values of the random vector (X,Y). b) Find the marginal pdf f2(y) of Y.
Suppose that a point X is selected at random from the interval (0,1). After the value X = x has been selected, a point Y is then chosen at random from the interval (0,x^2). a) Indicate the region R on...
3. Suppose a value is chosen "at random" in the interval [0,6]. In other words, r is an observed value of a random variable X U(0,6). The random variable X divides the interval [0,6] into two subintervals, the lengths of which are X and 6- X respectively. Denote by Y min(X, 6-X), the length of the shorter one of the two intervals. Find e probability PY> ) for any given y. Then find both the cdf and the pdf of...
For questions 5-8, consider the following experiment: Suppose the location of a particle in the plane is restricted to be within the region of the first quadrant enclosed by y = 0, y = 2, and 1-1 and that the z and y coorlinates of the point are d(scribed by the jointly continuous random variables X and Y, respectively, with joint pdf Uz, y) = cryl(0,1) (z)10.ェ2)(y) 5. Given this joint pdf, (a) Find the value of c that makes...
6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b) Obtain the marginal pdf of S. 6. Suppose that Xi,X2, X, is a random sample from the uniform distribution on (0,1). Let X(i), i = 1, , n denote the order statistics. (a) Obtain the joint distribution of R- X)-X) and SXXn/2 b)...
(7 points) Suppose X and Y are continuous random variables such that the pdf is f(x,y) xy with 0 sx s 1,0 s ys 1. a) Draw a graph that illustrates the domain of this pdf. b) Find the marginal pdfs of X and Y c) Compute μΧ, lly, σ' , σ' , Cov(X,Y),and ρ d) Determine the equation of the least squares regression line and draw it on your graph. (7 points) Suppose X and Y are continuous random...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
Suppose (X,Y ) is chosen according to the continuous uniform distribution on the triangle with vertices (0,0), (0,1) and (2,0), that is, the joint pdf of (X,Y ) is fX,Y (x,y) =c, for 0 ≤ x ≤ 2,0 ≤ y ≤ 1, 1/ 2x + y ≤ 1, 0 , else. (a) Find the value of c. (b) Calculate the pdf, the mean and variance of X. (c) Calculate the pdf and the mean of Y . (d) Calculate the...
Suppose that X is uniformly selected from the numbers 1, 2, 3 (that is, P(X = i) = 1/3 for i = 1, 2, or 3). Once X is selected, Y is chosen uniformly from the numbers 0, 1, ..., X. Find E[X | Y ]. (c) Compute EX Y 7. Suppose that X is uniformly selected from the numbers 1, 2, 3 (that is, P(X- i) 1/3 for ,X. Find i = 1,2, or 3). Once X is selected,...
9. Suppose a point (X,Y) is selected at random from inside the circle with radius 2 and center at (0,0). Find the joint p.d.f. of X and Y.
Q4. Suppose X is a random variables with possible values -1,1 and Y is another with possible values -1, 0,1. Suppose the joint probability distribution of X, Y, f(x, y), is given by: a. Find the marginal distribution of X and Y. b. Find E[Xly1] c. Compute Cov(X, Y) d. Are X, Y independent? justify e. Compute E(XYorl