Random variables 2345678910111 The lump in the middle indicates that sums close to 7 are the most likely. The total area of all the rectangles is 1 since the dice must take on exactly one of the sums in V {2,3 A closely-related idea is the cumulative distribution function(cdf) for a random vari ble R. This is a function CDFR: V-[0, 1]defined by CDFR(x)=Pr(R≤x) As an example, the cumulative distribution function for the random variable T is show below 1 CDFR() 1/2 x∈ The height of the i-th bar in the cumulative distribution function is equal to the sum of the heights of the leftmost i bars in the probability density function. This follows from the definitions of pdf and cdf CDFR(x)=Pr(R≤x) ∑Pr(R=y) PDF In summary, PDFR(a)measures the probability that R=x and CDFr(a)measur the probability that R s a. Both the PDFR and CDFR capture the same information� � Random Variables 7 6/36 6 PDFR(x) 3/36 - 2 3 4 5 6 7 8 9 10 11 12 x ∈ V The lump in the middle indicates that sums close to 7 are the most likely. The total area of all the rectangles is 1 since the dice must take on exactly one of the sums in V = {2, 3, . . . , 12}. A closely­related idea is the cumulative distribution function (cdf) for a random vari￾able R. This is a function CDFR : V → [0, 1] defined by: CDFR(x) = Pr (R ≤ x) As an example, the cumulative distribution function for the random variable T is shown below: 1 6 CDFR(x) 1/2 0 - 2 3 4 5 6 7 8 9 10 11 12 x ∈ V The height of the i­th bar in the cumulative distribution function is equal to the sum of the heights of the leftmost i bars in the probability density function. This follows from the definitions of pdf and cdf: CDFR(x) = Pr (R ≤ x) = Pr (R = y) y≤x = PDFR(y) y≤x In summary, PDFR(x) measures the probability that R = x and CDFR(x) measures the probability that R ≤ x. Both the PDFR and CDFR capture the same information