4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be...
Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ, by R2=X2+Y2,Θ = tan−1(Y/X).You can think of X and Y as the real and the imaginary part of a signal. Similarly, R2 is its power, Θ is the phase, and R is the magnitude of that signal. (a) Find the joint probability density function of R and Θ, i.e.,fR,Θ(r,θ).
Prove that Box-Muller method described in class generates independent standard normal random variables. 4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
Let X and Y be independent exponential random variables with parameter 1. Find the joint PDF of U and V. U = X + Y and V = X/(X + Y)
Exercise 7. Let X and Y be A. independent exponential random variables with a common parameter (1) Find the transform associated with aX +Y, where a is a constant. (2) Use the result of part (1) to find the PDF of aX +Y, for the case where a is positive and different than1 (3) Use the result of part (1) to find the PDF of X-Y. Justify your answers. Exercise 7. Let X and Y be A. independent exponential random...
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2). 3. The pair of random variables X and Y is uniformly distributed on the interior...
Suppose that random variables X and Y are independent. Further, X is an exponential random variable with parameter 1 = 3, and Y is an uniformly distributed random variable on the interval (0,4). Find the correlation between X and Y, rounded to nearest .xx
Problem 5 . This question considers uniform random points on the unit disc x2+92 〈 1 (a) A point (X, Y) is uniformly chosen in the unit disc. Find the CDF and PDF of its distance from the origin R X2 +Y2 (b) Compute the expected distance from the origin. (c) Determine the marginal PDF of X and Y (d) Are X and Y independent? (Justify your claims) e) One way to generate uniform random points on this disc is...
4. Let X and Y be independent exponential random variables with pa- rameter ? 1. Given that X and Y are independent, their joint pdf is given by the product of the individual pdfs of X and Y, that is, fxy(x,y) = fx(x)fy(y) The joint pdf is defined over the same set of r-values and y-values that the individual pdfs were defined for. Using this information, calculate P(X - Y < t) where you can assume t is a positive...
Let X and Y be independent random variables which are exponential with parameter lambda= 1, so then each has probability density function equal to f(x) = exp(-x) when x > 0, and zero otherwise. Compute the probability density function of X + Y . Show detailed explanations and reasoning for each step.