Question

CANONIMESTESI MÜHENDİSLİK FAKÜLTESİ FACULTY OF ENGINEERING 1. A shop produces pipes of 100cm length. But due to manufacturing errors, sometimes the pipes are too long or too short. Let X be the error in the length of pipes produced in a shop. X has the following probability distribution | (1 – x){if -1<x<1, otherwise (x)={419-** a. Find A. b. Find the cumulative distribution function of X? C. Find the expected value of X. d. Find the variance of X. What is the probability that a pipe is produced which is exactly 100cm long? f. What is the probability that the length of the pipe is less than 100.6 cm? e. I

Answer 1

And let y be the error in the length of pipes produced in a shop. The density of x.is if i << f(x) = { AC1= x2 otherwise cas Since fir) is a valid pol. Hence Splendx = 1 S'A11732 = A[-4-3° = [ 22-0] = $A so 8A = 1 -) A- mico (6) X is 2 х 3 So -ICX) (C) The cdf of F(x) = P(xsx) P(x5X) - s All-x)?d=ć = A [ 23 (1-2)3] F(X) = [ {8-0-413} The expectation of x is ELX) = 5 x f(x)dy s A x (1-x)2 dx EANS (66 +4?_2X2) dx] = A ( 12°1'+/47 2131) FAL0 +0-21*+)] = 2 *(-) = -2 The variance of is given by Vantx) = E(X2) - {E(X)32 E[x2] = $ '*? f(xdx= Ax2(1-x)? dx =Al ² + x4_2x5) dx = A Film +119.-2 POR] ++ ++)-14 })= x so Vanix) = -(-2)2- ] = 0+5 = 3 (d) So X 2 3 + 16 75 2 5 - 20

meam the =0 le A pipe is produced of length exactly too cm long. That error in length of pipes is zoro, il y=0 Hence the required probability is P(X=0) = (since x. is a continuous r. Ve , so the point probability is zero (f). If the length of pipe produced, as less than 100.6 cm that means the emos in length in less than 0.6 cm X <0.6. So the required prob is 0.6 0,6 07 Pl. X < 0.6);=0 flxidx = 1 A (1-x)?dx 3 8 11-0.6130 3 064 ) = 1 – 0.064 0.992, . [ - 0.064] - 8