Question

Let X denote the amount of space occupied by an article placed in a 1-ft³ packing container. The pdf of X is below.

f(x) =

90x8(1 − x)		0 < x < 1
0		otherwise

(a) Graph the pdf.

Obtain the cdf of X.

F(x) =

0 x < 0 0 ≤ x ≤ 1      1 x > 1

Graph the cdf of X.

(b) What is P(X ≤ 0.65) [i.e., F(0.65)]? (Round your answer to four decimal places.)


(c) Using the cdf from (a), what is P(0.4 < X ≤ 0.65)? (Round your answer to four decimal places.)


What is P(0.4 ≤ X ≤ 0.65)? (Round your answer to four decimal places.)


(d) What is the 75th percentile of the distribution? (Round your answer to four decimal places.)


(e) Compute E(X) and σ_X. (Round your answers to four decimal places.)

E(X)	=
σX	=

(f) What is the probability that X is more than 1 standard deviation from its mean value? (Round your answer to four decimal places.)

Answer 1

pdf 4 3 2 1 O — - 0.0 0.2 0.4 0.6 0.8 1.0

The pdf of X, 0<x<1 f(x): 90 x (1-x) floor 29-11-23 f(x) = Bera (19.2) osall - a 1992 niqob:107% Fix) = ax? (1-2 da 90sx & dx - qo Sa a che = 90/21T0. – 90%2Cyo a To = 10x9_900 = 29C10 - 90) : The colf of x x o F(x) = 2X? (10-9x) : 0LX<1

cdf 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0

® P ( x < 0.65)= F (0.65) - (0.65" (10 - 9x0.65) = 0.0859 © P(0.405*30.65) = F(0.65) - Folio) = [(0.65" (10-9x0.65) - [ (8.469 (10-9x0.40)] = 0.0843 @ Les x0.75 bee the 7th percentile 0.75= s f(x) dx 0.75= (10.75) (10 - q* *0.78) 3 x0.75 = 0.9036 28.75

© E(X) = Sx46dx - 1 1 - 3 dix Beta (9,2) Bela (102) Bela (9,2) To a 12 = 9 {(22) = Salfondx al S(1-x2 de Beta (9,2) Bela (112) Beta (9,2)

10xq 12 x 11 Var(x) = {(22) - {{(x2 ( 9) 9 (55-54 = 166 66

Hence, É (X) = 45 = 0.6818 fx = vaste) = a + 0.1113 Ô P(x- EX176x) - PC x > E62) +(x) = P(x>0.7932) = 1- F (0.7932) = { 0.3556 = 0.6444