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Consider the density function SAVE, 02, Y= 1); c) (C) P(X > Y);(d) P(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.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 1 2 3 I(x) 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) = - 01/2).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 f(x) = [3r, r>1, 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 2(n- 1) =0The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x sy, and assume that the joint density function of these variables is 12, 0