Q41Comprehension3 Marks4 Aug 2024PassageSuppose X=W+NX=W+NX=W+N, where WWW is discrete uniform on {−1,1}\{-1,1\}{−1,1} and N∼Normal(0,1)N\sim\mathrm{Normal}(0,1)N∼Normal(0,1) is independent of WWW. Based on the above data, answer the subquestions.What is the distribution of XXX?AfX(x)=12π[e−12(x+1)2+e−12(x−1)2]f_X(x)=\frac1{\sqrt{2\pi}}[e^{-\frac12(x+1)^2}+e^{-\frac12(x-1)^2}]fX(x)=2π1[e−21(x+1)2+e−21(x−1)2]BfX(x)=124π[e−(x+1)2/4+e−(x−1)2/4]f_X(x)=\frac1{2\sqrt{4\pi}}[e^{-(x+1)^2/4}+e^{-(x-1)^2/4}]fX(x)=24π1[e−(x+1)2/4+e−(x−1)2/4]CfX(x)=122π[e−12(x+1)2+e−12(x−1)2]f_X(x)=\frac1{2\sqrt{2\pi}}[e^{-\frac12(x+1)^2}+e^{-\frac12(x-1)^2}]fX(x)=22π1[e−21(x+1)2+e−21(x−1)2]DfX(x)=12πe−x2f_X(x)=\frac1{2\sqrt{\pi}}e^{-x^2}fX(x)=2π1e−x2
