Here is the probability density function of a random variable X with Gamma[2.5, 1.2] distribution: in32) = Clear (pdf, cdf, x] : {a, b} = {2.5, 1.2}; {Xlow, Xhigh) {0, Infinity); cdf[x] N[Evaluate [Gammacdf [x, a, b]]]; pdf [x_] D[cdf [x], x] = Out[36] 0.476882 x1.5 -0.833333 % The expected value: In 37] expect Integrate [x pdf [x], {x, Xlow, Xhigh}] = Out[371 3. The standard deviation: 10[38] standdev Sqrt(Integrate [ (x expect)^2 pdf [x], (x, xlow, Xhigh}]] Out[38= 1.89737 A plot of its probability density function: In39 practicalXhigh = expect + 5 standdev; pdfplot = plot [pdf [x] {x, xlow, practicalXhigh). PlotStyle -> ((Purple, Thicknes {expect, pdf [expect]}}]. AspectRatio -> 1] O(401 0.25 0.20 0.15 0.10 0.05 8 10 12 The x that lies at the bottom of the plotted line is the expected value of a Gamma[2.5, 1.2) random variable X Continue with this problem by finding the median and the mode of this distribution. Here is the probability density function of a random variable X with Gamma[2.5, 1.2] distribution: in32) = Clear (pdf, cdf, x] : {a, b} = {2.5, 1.2}; {Xlow, Xhigh) {0, Infinity); cdf[x] N[Evaluate [Gammacdf [x, a, b]]]; pdf [x_] D[cdf [x], x] = Out[36] 0.476882 x1.5 -0.833333 % The expected value: In 37] expect Integrate [x pdf [x], {x, Xlow, Xhigh}] = Out[371 3. The standard deviation: 10[38] standdev Sqrt(Integrate [ (x expect)^2 pdf [x], (x, xlow, Xhigh}]] Out[38= 1.89737 A plot of its probability density function: In39 practicalXhigh = expect + 5 standdev; pdfplot = plot [pdf [x] {x, xlow, practicalXhigh). PlotStyle -> ((Purple, Thicknes {expect, pdf [expect]}}]. AspectRatio -> 1] O(401 0.25 0.20 0.15 0.10 0.05 8 10 12 The x that lies at the bottom of the plotted line is the expected value of a Gamma[2.5, 1.2) random variable X Continue with this problem by finding the median and the mode of this distribution