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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.A continuous random variable X that can assume values between x = 1 and x = 3 has a density function given by f(x) = 1/2. (a) Show that the area under the curve is equal to 1. (b) Find P(2 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']