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

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one ifx13.

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 \\nnF(1) = \frac{(2 \times 1)-1}{(12)^2} = \frac{1}{(12)^2} = (\frac{[1]}{12})^2 \\nnX=2 \\nnF(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 \\nnX=3 \\nnF(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 \\nnF(4) = (\frac{[4]}{12})^2 \\nnX= 5 \\nnF(5) = (\frac{[5]}{12})^2 \\nnX= 6 \\nnF(6) = (\frac{[6]}{12})^2 \\nnu2026 \\nnX= 12 \\nnF(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}nn0\;;x\leqslant 0 & \\nn([x]/12)^2 \; ;0<x<13 & \\nn1 \; ; x \geqslant 13 &nn\end{matrix}\right.”

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