Question

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 f(x)= SA(1 – x)2 if –1<x<1, 10 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? What is the probability that the length of the pipe is less than 100.6 cm? f.

Answer 1

-1 < 1 cl (w) - A(1-742) flur '6 O Otherwise entire ۵۱ Since Sum y poobabiling distribution over the is 1 I we i have sange All-712) ='1 04 Í A (1-412) => -> A S1 -> megrering | * -st). ***)-(-5) [ + ] (3) A A = 4 A ( b) Now n∞ cumulative distributive function is given by : 1 * gets dt fx la) we have - tai f(t) = (1-4). Otherwise . 0 Now Ex (2) (1-t²) dt- integrating.

V-V 2.(*)-(-- al 1) 3 (W)- 23 - XX +3) برما کے ~ 3 9x-303 + 12 12 Therm fore, Ex (a) iſ *'s-1 et sherb -1231 K> 1 is given by (c) Expected value of px pins de E (x) - - {(x) 2. § (1-2) dx 41 (x)3 sw nous da ECY) 11 [Cartoo-,) -(7-5) 3 [ 23 2 [02:")- (2:12] ECX) -

x is Expected value Therefore Zero of x is (a) now vorrance of given by : (X)n E Cx+) - ECx2 Now {cat) - [xfinan {(x²) 22 23 (-4) dx ECx2) x²_x da X **/' *-*" de 3 {(x2) -1 کر۱-) ک 1 ElX) 3 ㅗ s [( ។ ) - (0 count] i (83)-( 3 *$)] 11 {(x²) g (X)3 [(E-). 1 ECx2) 11 ( is) (7*)3 al Therefore vaalance is Sluce ECX) = 0 CX) Ecx4) ut 5