{Solved R Exercise:} Section 2.3, # 4 The following two questions pertain to the spray tank irrigator of Exercise 3 of Section 2.3. Part a: The water PH level is measured over the next two irrigations Part i: Give the sample space of this experiment, and its size, using R commands t=seq(6.8, 7.2, 0.1) G-expand. grid(x1-t, x2-t *# 2 6.9 6.8 *# 3 7 :1 6 .8 26.8 6 7.2 7.2 length (G) ## [1] 2 The rows of G give realizations of the realizations of two irrigations. Part (ii): Give the sample space of the experiment that records the average of the two commands PH measurements, and the corresponding probability mass function, using the additional R table( (X1+X2)/2) /length (X1) ## 6.8 6.85 6.9 6.95 7 7.05 7.1 7.15 7.2 ## 0.04 0.08 0.12 0.16 0.20 0.16 0.12 0.08 0.04 Part (b): Using R commands similar to the above, give the probability mass function of irrigations. the experiment that records the average of the pH measurements taken over the next five In the R code below, we do not print out G because it is too big. Instead we print out the dimensions of G and the first/last six rows. t seq(6.8, 7.2, 0.1) H-expand. grid (X1-t, X2-t, X3-t, X4-t, x5-t) dim (H) ## [1] 3125 head (H) X1 X2 X3 X4 X5 ## 1 6.8 6.8 6.8 6.8 6.8 2 6.9 6.8 6.8 6.8 6.8 *# 3 7.0 6.8 6.8 6.8 6.8 ## 4 7.1 6.8 6.8 6.8 6.8 ## 5 7.2 6.8 6.8 6.8 6.8 ## 6 6.8 6.9 6.8 6.8 6.8 tail (H) *# X1 X2 X3 X4 X ## 3120 7.2 7.1 7.2 7.2 7.2 8 7.2 7.2 7.2 7.2 ## 3122 6.9 7.2 7.2 7.2 7.2 # # 3124 7:1 7.2 72 72 73 ## 3125 7.2 7.2 7.2 7.2 7.2 length (H$X1) *# [1] 3125 In the next code chunk we compute the PMF function of the average PH of 5 irrigations attach (H) ## The following objects are masked from G: f = table ((X1+X2+X3+X4+X5)/5) /length(X1) pmf 6.8 6.82 *# 0.00032 0.00160 0. 00480 0. 01120 0.02240 0.03872 0.05920 0.08160 0. 10240 0.11680 6.9 6.92 7.02 7.08 *# 0.12192 0.11680 0. 10240 0. 08160 0.05920 0.03872 0.02240 0.01120 0.00480 0.00160 7.12 7.2 ## 0.00032 barplot (pmf , cex. names - 0.5) 0.12 0.08 0.04 0.00 68 6.82 684 6.85 6.28 6.9 6.92 694 696 6.98 7 7.02 7.04 7.05 7.08 7.1 7.12 7.14 7.16 7.18 72 In the above example the argument "cer.names = 0.5" is needed to reduce the size of the bar name labels so that they are each visible