Question

10pts. 7875 Problem No. 6.3 Elongation (in %) of steel plates treated with aluminum are random with probability density function 15 sxs 30 True (Note: "True" means "Otherwise" or "Elsewere") 1) What proportion of steel plates have elongations greater than 25%? 2) Find the mean elongation. 3) Find the variance of the elongations. 4) Find the standard deviation of the elongations. 5) Find the cumulative distribution function of the elongations. 6) A particular plate elongates 20%. What proportion of plates elongate more than this? Show your explanations Displaying only the final answer is not enough to get credit Note: round calculated numerical values to the fourth decimal place where applicable. Solution (Show all intermediate steps, formulas, calculations, explanations and comments below this time. Doe't write above this line)

Answer 1

☺ sol Giren problem is 2 X f(x) = 15 sx = 30 7875 Ow 1) P(x>25) = f(x) dx 25 30 S x2 dz 7875 25 30 x² da 7875 25 30 (*) 125 7875 (3073 - 2533 = 7875 -0.481481 0.4815 26.48 0.4815 proportion of steel plates have elongations greater 25%. (Any) than 2) Mean elongation E (K) = elx = fx f(a) do 2 30 2² ef de 7875

I 2 ap{t 7875 30 Z 124 (三) Bài 2. () 7875 ا €(x) = 4x a isthit olhza 24. 1071 Mean elongation = 24.1071 (Ang) 3) Find vasience of the elongation V6 € (**) - (ER))? E (x)² = 8x²f 122 de 30 x²x2 dae 2 - S 7875 is 30 z 1 [ary 7975 15 - [6056 -cos N 7875 5 597.8571429 ~/597-8571 V(x) = (5 97.8571) - (24.107172 06.2008 Variance of the elongation =16.704829 / 16.7048 in 16.9048 ()

u standard deviation 3 SD (X) Jvcx) 126.7048 =4.08 7150597 ~ 4.0872 SOLA = 4.0872 standard deviation of elongation is 4.0872 (Any) 5) Cumulative distribution function F(x) x cls GO then F(x) = f f (t) dt=so dt = 0 Go 15 = x 6 30 15 х 2 F(x)= ا s odtt a dt 7875 D zot 7875 ㅗ 7875 12 [*'s?) (x2-3375] ! 2 23625 x3_3375 2 23625

If x=30 2 30 f(x) = s odt & S f (t) dt tS odt 30 15 2 I Rotito Cumulative distribution function is et Xcit -3335 f(x)= If 155X230 23.625 SA X 330 6) here fand P (x320) = 1-P (x = 20) zl.f(20) (2003 - 3375 21- 23625 Claro [200 2000-3375 2 12 23625 20-804232 Or8042 (Ars