The Normal PDF and CDFLab Assignment # 3Due DateOutlineThe Normal PDF and CDFLab Assignment # 3Lab3: Continuous DistributionsM. George AkritasM. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3The Normal PDF and CDFLab Assignment # 3Due DateM. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Plotting and comparing1. Plotting the standard normal pdf and cdf:1.1 Make a grid of points on the x-axis: ”x=seq(-4,4,0.01)”1.2 Calculate the standard normal pdf and cdf for each point inthe grid: ”yd=dnorm(x), yp=pnorm(x)” (comma within ” ”means hit return).1.3 Plot the pdf and cdf: ”plot(x,yd,type=”l”),plot(x,yp,type=”l”)”1.4 Superimpose the two plots: ”plot(x,yd,type=”l”),lines(x,yp,type=”l”)”2. Superimposing three normal PDFs:”plot(x,dnorm(x,0,0.5),type=”l”,col=”blue”),lines(x,dnorm(x),type=”l”, col=”red”),lines(x,dnorm(x,0,2),type=”l”,col=”black”)”M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Marking and shading1. Plot the standard normal PDF and mark the 90th percentile:”plot(x, dnorm(x), type=”l”),lines(qnorm(0.9),dnorm(qnorm(0.9)), type=”h”, col=”red”)”2. Mark the 90th percentile on the CDF plot: ”plot(x, pnorm(x),type=”l”), lines(qnorm(0.9),pnorm(qnorm(0.9)), type=”h”,col=”red”)”3. Shade the area under the N(0,1) pdf to the right of the 90thpercentile: ”x1=seq(qnorm(0.9),4,0.01), y1=dnorm(x1),plot(x,dnorm(x),type=”l”), lines(x1,y1,type=”h”,col=”red”)”M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Due DateProperties of the Normal1. Using the R commands ”pnorm(-x)” for Φ(−x ), and ”1−pnorm(x)” for 1 − Φ(x ), and three positive x -values of yourchoice (it is best if your numbers are not greater than 3)verify that Φ(−x ) = 1 − Φ(x ) holds for any x > 0.IReport the three x values that you chose and thecorresponding Φ(−x ) and 1 − Φ(x) values.IGive an explanation of Φ(−x) = 1 − Φ(x) using thegeometrical interpretation of these quantities as areas underthe standard normal pdf.A non-geometrical explanation is as follows:If Z ∼ N(0, 1) then −Z ∼ N(0, 1). Thus, for any x > 0Φ(−x ) = P (Z ≤ −x ) = P(−Z > x ) = P(Z > x) = 1 − Φ(x )M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Due Date2. Using the R commands ”pnorm(x,µ, σ)” for P (X ≤ x), whereX ∼ N (µ, σ2), and ”pnorm((x − µ)/σ)” for Φ((x − µ)/σ),and three sets of (x , µ, σ) values of your choice (pick anynumber for µ, any positive number for σ, and pick a value forx such that µ − 3σ < x < µ + 3σ, and do this three times)verify thatP(X ≤ x ) = Φx − µσIReport the three sets of (x, µ, σ) values that you chose and thecorresponding P(X ≤ x ) and Φx −µσvalues.M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Due Date3. Using the R command ”qnorm(1 − α,µ, σ)” for the100(1 − α)th percentile, xα, of X ∼ N (µ, σ2), verify thatx1−α= µ − (xα− µ)by computing the two sides of the above equation for threedifferent values of α between 0 and 0.5 and any pair of (µ, σ)values.IReport the pair of (µ, σ) values you chose, the three differentvalues of α you chose, and the corresponding values of the twosides of the equation.M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Due DateEmail your answers to Ms. Lauren Kraus by Friday, Oct. 7th.M. George Akritas Lab3: Continuous DistributionsOutlineThe Normal PDF and CDFLab Assignment # 3Due DateIGo to previous lab http://www.stat.psu.edu/~mga/401/course.info/b.lab-Ch2.pdfIGo to next lab http://www.stat.psu.edu/~mga/401/course.info/b.lab-Ch4a.pdfIGo to the Stat 401 home pagehttp://www.stat.psu.edu/~mga/401/course.info/Ihttp://www.stat.psu.edu/~mgaIhttp://www.google.comM. George Akritas Lab3: Continuous