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

x 3 2t² + 28 3 2 -2 3 28 26 3 28 [t - [*-**+ **)-(2-4-3) [x (1-x+ 22 ) = *(-x+*) + 23 cdf Fy (x) = 30 (1-x+ 24 + 1 4 @ Expected value of x E67) = }<f(z) dx :. () = }4.3 (1-2)*dze (using calculator) X -2 - 2 8 -- 1.19285 7 Variance of a Noul E(x²) v(x) = E(x2) - E²(x) -(1-2) ²dx 2 3 28 - 2 68 = 35 68 ooit 156 245 -0.63673 : v(x) = 35

علي O error. Noul error Nous to have pipe exactly bong, exactly 100cm must have exactly follows continuous distribution hence P(x=a)=0 & aeth :. P(getting 0) hence the probability that the pipe is exactly 100cm long is o. ↑ Nous to find the probability that length of the pipe must be less than 101cm, less than find P(x1) = f(a) doc - Ę (0) must have ve error So we -2 IĘC = 3x1 28 (1-1 + 2) + 11 4 x š z 28 + 13 14 = 13 27 28 19 EC:) = x + An). The ng required probability is 27 28