solve them

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.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 total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function 0<: f elsewhere. find the probability that over a period of one year family runs their vacuum cleaner less than hours between and hours.magnetron tubes are produced on an automated assembly line. sampling plan is used periodically to assess quality lengths tubes. this measurement subject uncertainty. it thought random tube meets length specification in which measured. show function y number out meet given by following discrete function: suppose selections made off line outside specifications. use fly above either support or refute conjecture single specifications.based extensive testing determined manufacturer washing machine time years before major repair required characterized density critics would certainly consider product bargain if unlikely require sixth year. comment determining> 6). (b) What is the probability that a major repair occurs in the first year?\fIn a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each. The data are as follows: Company A: 9.3 8.8 6.8 8.7 8.5 6.7 8.06.59.27.0 Company B: 11.0 9.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6 (a) Calculate the sample mean and median for the data for the two companies. (b) Plot the data for the two companies on the same line and give your impression regarding any apparent differences between the two companies. An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 3 of these automobiles at random, list the elements of the sample space S, using the letters Band / for blemished and nonblemished, respectively; then to each sample point assign a value x of the random variable X representing the number of automobiles with paint blemishes purchased by the agency.Three cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades. Find a formula for the probability distribution of the random variable X representing the outcome when a single die is rolled once. Find the probability distribution of the random variable Win Exercise 3.3, assuming that the coin is biased so that a head is twice as likely to occur as a tail. Reference: Exercise 3.3: Let Wbe a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W