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- A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.
- If two random variables X1 and X2 have the joint density function given by f (x1, x2) = x1x2, 0 < x1 < 1, 0 < x2 < 2 0, otherwise Find the probability that (a) Both random variables will take on values less than 1 (b) The sum of the values taken on by the two random variables will be less than 1.A random variable X has a Pareto distribution if andonly if its probability density is given by f(x) =⎧⎪⎨⎪⎩αxα+1 for x > 10 elsewhere where α > 0. Show that μ r exists only if r < α.Consider two random variables X and Y whose joint probability density function is given byf_X,Y (x, y) = c if x + y ≤ 1, x ≤ 1, and y ≤ 1,0 otherwise What is the value of c?
- A continuous random variable has the probability density function f(x) = 1/8 for 0 ≤ x ≤ 8. What is the expected value of X?If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.
- Suppose that X is a continuous random variable with density function f(x). If f(x)=k for −5≤x≤3 and f(x)=0 otherwise, determine the value of k.Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.Suppose that the random variable B has the standard normal density. What is the conditional probability density function of the sum of the two roots of the quadratic equation x2 + 2Bx + 1 = 0 given that the two roots are real? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTION