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 90x8(1-x) 0 0<x<1 otherwise rx) = Adapt the following R code to graph the PDF in R. here the pdf is fx)-ax*u-x) 0<x<1 otherwise ### 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, pdf(r), type-xlabmainContinuous VAR, lwd2) # # # End Code (a) Choose the correct graph of the pdf f(x) f(x) 3 3 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 f(x) f(x) 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 (b)Obtain the cdf of X. x9(10-9x) F(x) - x1 (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 F(x) 1.0 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.2 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 F(x) 1.0 F(x) 1.0 0.8 0.8 0.6 0.6 0.4 0.2 0.4 0.2 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 (d) What is PX S 0.65) [i.e., F(0.65)]? (Round your answer to four decimal places.) 08595 (e) Using the cdf from (b), what is P(0.35 s X s 0.65)? (Round your answer to four decimal places.) 08548 (f) What is P(0.35 X 0.55)? (Round your answer to four decimal places.) 02271 (g) What is the 75th percentile of the distribution? (Round your answer to four decimal places.) 9036 (h) Compute the average and standard deviation of X. (Round your answers to four decimal places.) (i) What is the probability that X is within 1 standard deviation from its mean value? Hint: Find the P(Hx-Ox S X S xtox) (Round your answer to four decimal places.) 06344X Show My Work (Optional)I can't find the solution for (i), I tried the hint but still lost  

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Answer #1

(i) Correct answer: 0.6854461

since

Console Terminal > a-90 > b-8 pdf-function(x) a (xAb) (1-x) >ub-0. 8181+0.1112 1b-0.8181-0.1112 as.numeric(integrate(pdf,0,ub) [1])-as. numeric(integrate(pdf,0,1b)[1]) [1] 0. 6854461

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