 Solution

ID: 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/solution

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

1/ R

from

s = 0

r

, followed by zero from

r

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.

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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