Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 14 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii
- iii a
  - Solution iii

i

Solution

A displayed minute contains one minute of possible start times, so the specified uniform distribution has probability density function

p (s) = {1, 0, 0 \leq s < 1, otherwise .

Endpoint choices for a probability density function do not affect any probabilities.

ii

Solution

The graph is a rectangle of height one over

[0, 1)

, with zero height elsewhere. Its unit area is the required normalization of the probability density function.

Figure 1.
Uniform endpoint uncertainty and triangular elapsed-time uncertainty
. Left: the uniform distribution of one endpoint offset. Right: the triangular distribution of the difference of two independent endpoint offsets. Shading marks the event $T > 1/2$ . Details.

iii

Solution

For the uniform distribution, the expected value and second moment are

E [S] = \int_{0}^{1} s d s = \frac{1}{2}, E [S^{2}] = \int_{0}^{1} s^{2} d s = \frac{1}{3} .

Hence the variance is

E [S] = \frac{1}{2} minute, Var (S) = \frac{1}{3} - \frac{1}{4} = \frac{1}{12} minute^{2} .

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version