Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 16 / a / iii

Table of contents
- Solution iii

Solution

The graph of the positive-delay probability density function is a horizontal segment at height

1/ R

from

s = 0

to

r

, followed by zero from

r

to

R

. Values at individual endpoints do not affect probabilities. The separate atom

1 - r / R

at zero is not a finite-height part of this density graph.

Figure 1.
Positive-delay density and the full cumulative distribution, whose jump records the no-delay atom
. Details.

The companion cumulative distribution function in the figure is

F_{S} (s) = 0

for

s < 0

,

F_{S} (s) = 1 - r / R + s / R

for

0 \leq s < r

, and

F_{S} (s) = 1

for

s \geq r

. Its jump at zero displays the discrete part without confusing density height with probability mass.

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