5. Let X be uniformly distributed over (0,1). a) Find the density function of Y =...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
Let there be U, a random variable that is uniformly distributed over [0,1] . Find: 1) Density function of the random variable Y=min{U,1-U}. How is Y distributed? 2) Density function of 2Y 3)E(Y) and Var(Y) U Uni0,1
Let X be a uniformly distributed random variable on [0,1]. Then, X divides [0,1] into the subintervals [0,X] and [x,1]. By symmetry, each subinterval has a mean length 0.5. Now pick one of the subintervals at random in the following way: Let Y be independent of X and uniformly distributed on [0,1], and pick the subinterval [0,X], or (X,1] that Y falls in. Let L be the length of the subinterval so chosen. What is the mean length of L...
5. Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices (0,0), (2,0),(1,1), (0,1) Uniformly distributed means that the joint probability density function of X and Y is a constant on D (equal to 1/area(D)). (a) Do you think Cov(X, Y) is positive, negative, or zero? Can you answer this without doing any calculations? (b) Compute Cov(X, Y) and pxyCorr(X, Y)
Let X, Y be iid random variables that are both uniformly distributed over the interval (0,1). Let U = X/Y. Calculate both the CDF and the pdf of U, and draw graphs of both functions.
If X is uniformly distributed over (0, 2), find the density function of Y = e X. The density can be given only on the interval (1, e 2 ) where it is non-zero.
Consider two independent random variables X1 and X2. (continuous) uniformly distributed over (0,1). Let Y by the maximum of the two random variables with cumulative distribution function Fy(y). Find Fy (y) where y=0.9. Show all work solution = 0.81
Let X be a uniform random variable over (0,1). Let a and b be two positive numbers and let Y = aX+b. (a) Determine the moment generating function of X. (b) Determine the moment generating function of Y. (c) Using the moment generating function of Y, show that Y is uniformly distributed over an interval(a, a+b).
Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X,Y) and the marginal density functions of X and Y. (b) Find E[X] and E[Y]. (c) Are X and Y independent?
Let X be Uniformly distributed over the interval [0,π/2][0,π/2]. Find the density function for Y=sinXY=sinX. Evaluate the density function (to 2 d.p.) at the value 0.1. the density function is ?