Answer to Question #240032 in Statistics and Probability for Rayen

Question #240032

Denote by [x] the greatest integer not exceeding x. For the pdf in Example 2.2.1, show

that the CDF can be represented as F(x) = ([x]/12)2 for O < z < 13, zero if x O, and

one ifx13.

Expert's answer

Let [x] be the greatest integer not exceeding x.

The probability density function in Example 2.2.1 is given as follows:

"f(x) = \\frac{2x-1}{12^2}"

for x=1,2,…,12

The cumulative distribution function (CDF) of a random variable X is defined for any real x is,

F(x) =P(X≤x)

Calculate the cumulative distribution function by using the following table:

"X=1 \\\\\n\nF(1) = \\frac{(2 \\times 1)-1}{(12)^2} = \\frac{1}{(12)^2} = (\\frac{[1]}{12})^2 \\\\\n\nX=2 \\\\\n\nF(2) = \\frac{(2 \\times 1)-1}{(12)^2} + \\frac{(2 \\times 2)-1}{(12)^2}= \\frac{1}{(12)^2} + \\frac{3}{(12)^2}= (\\frac{[2]}{12})^2 \\\\\n\nX=3 \\\\\n\nF(3) = \\frac{(2 \\times 1)-1}{(12)^2} + \\frac{(2 \\times 2)-1}{(12)^2} + \\frac{(2 \\times 3)-1}{(12)^2}= \\frac{1}{(12)^2} + \\frac{3}{(12)^2} + \\frac{5}{(12)^2}= (\\frac{[3]}{12})^2"

Similarly, for the remaining terms, the cumulative distribution function is obtained as,

"X= 4 \\\\\n\nF(4) = (\\frac{[4]}{12})^2 \\\\\n\nX= 5 \\\\\n\nF(5) = (\\frac{[5]}{12})^2 \\\\\n\nX= 6 \\\\\n\nF(6) = (\\frac{[6]}{12})^2 \\\\\n\n\u2026 \\\\\n\nX= 12 \\\\\n\nF(12) = (\\frac{[12]}{12})^2"

By the above equations, the cumulative distribution function is represented as,

"F(x) = (\\frac{[x]}{12})^2"

Therefore, the cumulative distribution function is,

"F(x)= \\left\\{\\begin{matrix}\n\n0\\;;x\\leqslant 0 & \\\\\n\n([x]\/12)^2 \\; ;0<x<13 & \\\\\n\n1 \\; ; x \\geqslant 13 &\n\n\\end{matrix}\\right."