Problem 7. Suppose that a particle is fired from the origin in the (x, y) plane in a straight line at a random angle where is chosen uniformly from the interval (-). Consider the random variable Y= y-coordinate of the intersection between the point and the vertical line x = 1. Compute the comulative density function for Y, and conclude that the PDF is given by: 1 fy(y): = π(1+ y²)
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- Suppose you have two independent random variables X ~ Y ~ Exponential(A2), A1 > 0, d2 > 0. Find: Exponential(A1) and (a) The joint density function for (X,Y). (b) P( > 1). (c) P(X +Y > 1).Let (X,Y) is a two-dimensional random variable with the pdf f(x,y) = {* (x + y _02. Let X be a Chi-squared random variable with density function fx(x) = exp (-;), a x > 0. Suppose Y is independent of, and has the same distribution as, X. (a) Let Z = VX. Show that the density function of Z is given by Sale) = 2a exp (-) z > 0. Hence state the value of a.Let Yi,..., Yn denote a random sample from the density function given by fy (ul0) = ry"-le-"10, for 0 > 0, y > 0. where r is a positive constant. Find the MLE of 0. Is the estimator an MVUE for 0?The density function, ху fy(x, y) = f(x) =9 0, 0b) Let Z₁ = X-XN (0,1), and W₁ dx YHY~N(0,1), for i = 1,2,3,...,10, then: dy i) State, with parameter(s), the probability distribution of the statistic, T = - 54 1² ii) Find the mean and variance of the statistic T = Σ},wp? Σ1,2,3 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ẞ such that P(T> B) = 0.01, where T = Σ₁Z₁² + ₁ W₁².let X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.6) The random variables X and Y have joint density f( x, y) = x2-y2 for 1< x, 1s y and f( x, y) = 0 otherwise. %3! Compute the joint density of U = X /Y and V = X Y.A random variable Y have a distribution with parameter a > 0 and y, > 0 if its density function is given by: (ay“ if y> yo fV) = ya+T (0, elsewhere i) Derive the mean of Y. Show all necessary steps. ay (a-2)(a-1)²' ii) Show whether or not that the variance of Y is ·A projectile is launched at an angle theta with respect to the surface with velocity v0 (deterministic). If the angle of inclination is a uniform random variable in [0, pi/2 ], calculate the distribution function of the variable R defined as the point of impact of the projectile on the ground, measured from the origin. Also calculate your expected valueConsider a random sample X1,..., X, from the distribution with density f(x; 0) = e-(a-0), r> 0 and parameter 0 > 0. Hint: you may find it useful to show that this density is in fact the density of a random variable Y + 0, where Y is Exponential with parameter 1. (1) Find the method of moment estimator ô of 0. Show that ô is unbiased and consistent, and calculate its MSE. (2) Prove that the MLE Õ of 0 is X(1) (the 1st order statistic). (3) Calculate the bias and MSE of ő and prove that ô is consistent. (4) Which estimator do you prefer, ô or ô? Provide a brief explanation of your answer.SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON