Question

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 if -2<x<2, 0 otherwise a. Find A. b. Find the cumulative distribution function of X? c. Find the expected value of X. d. Find the variance of X. e. 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 101cm?

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Abdul-Rahim Taysir

Answer 1

Given that o E AC1-22 ; if a 2<x<2 flores- ; otherwise, a) To find we know that if fear is a pdf, s from dx = pdf, then 2 > A(1-x)2 de a - 2 2 A S(1-28 +223d244 v - 2 2 НА =1 X-X² + x 3 > A [(2 - 4 + 8 ) (-2-4-43 = 28 A =) 3 AS 28

(b) To find cumulative of X. distribution function (2 2 1. e. F(x) = P(xca)= Stranda 3 (1-x2 da 28 2 3 2x+x2) dx 28 3 x-x2 + x3 28 3 28 14***) - (+376 [(38-3 3 26 34-382+83) 28 26+(38-342+3) 28 Thus, if Xe-2 F(X) = x3 3x2+3 8+26 if -2<x<2 28 if x72 -

c) To find F(x) i.e. expected value of x.. F(X) = Seferes da 2 2 드 ( x. 2 (1.882 da 25 Sx(1-2x+2²) de { x = 2x² +28) iz - 2 3 28 - 2 2 2 3 28 [ 203 t RADIO 2 3 3 (+49-(2+1+99 28 -32 3 28 E(X) =

d) To find variance of a i.e, V(X). V(X) = f(x4- (EX))? * consider 2 E(x²) = (xe. fex de x2 (1-x22 de - 2 2 sxr (1-28+82) dx 2 12.2 x²_2x3 + x4) dx 2. 2x4 3 E(X2) 272 This using *) we get (= 3² 272 15 since F(X) = 8 7 V(X) = 12368 = 16.8272 735 This VOXDE 12368 735