ex.3 pg289
let \(X_1\) and \(X_2\) be independent random variables with common distribution \[px = (\begin{matrix} 0 & 1 & 2\\1/8&3/8&1/2 \end{matrix})\] find the distrubution of the sum \(X_1+X_2.\)
let \(P_z = P(X_1 + X_2)\)
px1_0 = px2_0 = 1/8
px1_1 = px2_1 = 3/8
px1_2 = px2_2 = 1/2\(P_z(0) = P_{X_1}(0)*P_{X_2}(0)\)
pz_0 = px1_0*px2_0
pz_0## [1] 0.015625\(P_z(1) = P_{X_1}(0)*P_{X_2}(1) + P_{X_1}(1)*P_{X_2}(0)\)
pz_1 = px1_0*px2_1 + px1_1*px2_0
pz_1## [1] 0.09375\(P_z(2) = P_{X_1}(0)*P_{X_2}(2) + P_{X_1}(1)*P_{X_2}(1)+P_{X_1}(2)*P_{X_2}(0)\)
pz_2 = px1_0*px2_2 + px1_1*px2_1 + px1_2*px2_0
pz_2## [1] 0.265625\(P_z(3) = P_{X_1}(1)*P_{X_2}(2) + P_{X_1}(2)*P_{X_2}(1)\)
pz_3 = px1_1*px2_2 + px1_2*px2_1
pz_3## [1] 0.375\(P_z(4) = P_{X_1}(2)*P_{X_2}(2)\)
pz_4 = px1_2*px2_2
pz_4## [1] 0.25# check the sum of pz
sum(pz_0,pz_1,pz_2,pz_3,pz_4)## [1] 1plot distribution
library(ggplot2)
data = c(pz_0,pz_1,pz_2,pz_3,pz_4)
barplot(data,xlim = c(0,6), ylab = "probability",xlab = 'pz', main = "the distribution of pz")