Given the values P(t) of the cumulative distribution function at t = 0, 5, 10, ..., 30, we sketch a histogram representing the proportions of occurrences in each of the t ranges.

We see that

• the probability of an observation between t = 0 and 5 is the difference .03 between the cumulative probabilities at t = 0 and t = 5,
• the probability of an observation between t = 5 and 10 is the difference .05 between the cumulative probabilities at t = 5 and t = 10,
• the probability of an observation between t = 10 and 15 is the difference .13 between the cumulative probabilities at t = 10 and t = 15,
• the probability of an observation between t = 15 and 20 is the difference .17 between the cumulative probabilities at t = 15 and t = 20,
• the probability of an observation between t = 20 and 25 is the difference .42 between the cumulative probabilities at t = 20 and t = 25,
• the probability of an observation between t = 25 and 30 is the difference .18 between the cumulative probabilities at t = 25 and t = 30.

If we had more detailed knowledge of the cumulative probability distribution, we could sketch a more refined histogram, which would probably approach the distribution sketched in the graph at lower left in the figure below.

• The peak value would probably occur somewhere in the range from t = 20 to t = 25.
• The corresponding cumulative distribution function would be the antiderivative function, which would have its greatest slope where p(t) has its greatest values, probably in the t = 20 to t = 25 range.

Video Clip #01

The relationship between the probability distribution function p(t) and cumulative distribution function P(t) is depicted below.

• The probability p(2 < t < 4) of occurrence between t = 2 and t = 4 is represented by the indicated area under the graph of p(t).
• The same probability is by the Fundamental Theorem of Calculus the difference of the values P(2) and P(4) of the antiderivative, the cumulative distribution function P(t).

You should sketch this distribution function for practice and for reference in what follows.

• The probability that x lies between 0 and 1 is p( 0 <= x <= 1), and is the area under the p(x) curve between x = 0 and x = 1.
• We represent this area as an integral and see that it is equal to 1/32.

To find the average value we calculate the first moment, obtaining xAve = 320 / 192 (approximately 1.67).'

• If we were to add up a large number of occurrences governed by this distribution, we would be very likely find that their average would be close to 1.67.

To find the median we set the integral representing the area under the curve from 0 to xMedian equal to 1/2.

• If we evaluate the integral we will see that xMed ^ 5 / 32 = 1/2, so xMed = 16^(1/5) = 1.74 (approx.).
• This means that half the values will tend to lie at x values < 1.74, with half f(x) values > 1/74.