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Let X denote the amount of space occupied by an article placed in a 1-ft3 packing container. The pdf of X is below. 72x(1-x) 0 0<x<1 otherwise f(x) = Adapt the following R code to graph the PDF in R. axb(1-x) 0 < x < 1 otherwise where the pdf is f(x) = ### R Code a-a ; b-b , # # # You must plug in values for a and b. r-seq(0, 1,0.01) # Defines range of X from 0 to 1 pdf = function(x)(aX-b*(1-x)) # Creates the pdf function plot(r pdfCr), type-, xlab -mainContinuous VAR, lwd 2) # # # End Code

(a) Choose the correct graph of the pdf. f(x) f(x) 0.2 0.4 0.608 1.0x 1.0 f(x) f(x) 0.2 0.4 0.608 1.0- 0.204 0.6 0.8 1.0 (bjobtain the cdf of X. F(x) -

(c) Graph the cdf of X. Adapt the R code from the pdf plot to graph the cdf according to your calculated function for F(X). F(x) 1.0 0.8 F(x) 1.0 0.8 0.6 0.4 0.2 0.6 0.4 0.2 0.2 0.4 0.6 08 1.0 0.2 0.4 0.6 08 1.0 F(x) 1.0 0.8 0.6 0.4 F(x) 1.0 0.8 0.6 0.4 0.2 0.6 0.8 10 02 0.4 0.2 0.4 0.6 0.8 1.0
d) What is PX s 0.7) [ie, FO.)1? (Round your answer to four dedimal places.) (e) Using the cdf from (b), what is P(0.45 s Xs 0.7)? (Round your answer to four decimal places.) (0) What is P(0.45 s X s 0.8)? (Round your answer to four decimal places.) (o) What is the 75th percentile of the distribution? (Round your answer to four decimal places.) (h) Compute the average and standard deviation of X. (Round your answers to four decimal places.) #x-SD(X)- what is the pr babity that x is within 1 standard deviation from its mean value? Hint Find the PVrơx SX μ Round your answer to four decimal places
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