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Consider the density function JKVI, 02, Y = 1); c) (C) PIX > Y);(d) F(X + Y = 4). A coin is flipped until 3 heads in succession occur. List only those elements of the sample space that require 6 or less tosses. Is this a discrete sample space? Explain.For the density function of Exercise 3.17, find F(x). Use it to evaluate P(2 2).The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by 0 1 2 3 4 (C) 0.41 0.37 0.16 0.05 0.01 Construct the cumulative distribution function of X. Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density f (z, y) = Ocr1/2).An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by (3r , :>1, f(z) = 10. elsewhere. (a) Verify that this is a valid density function. (b) Evaluate F(x). (c) What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers? Show that the n pieces of information in _ (x -") are not independent; that is, show that S(n-1)=0The proportion of people who respond to a certain mailorder solicitation is a continuous random variable X that has the density function \"1}: Elf'1, cxcl, I]. elsewhere. {a} Showmat F']