Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 14 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

Subtracting the constant translates the event without changing its probability. Therefore

P (a \leq U \leq b) = P (a - c \leq X \leq b - c) = \int_{a - c}^{b - c} f (x) d x .

This is a translation of the interval in the probability distribution of

X

.

ii

Solution

Apply the preceding interval formula on a short interval and use the change of variables

v = x + c

:

P (u \leq U \leq u + d u) = \int_{u - c}^{u + d u - c} f (x) d x = \int_{u}^{u + d u} f (v - c) d v .

Comparison with the defining integral for a probability density gives

t (u) = f (u - c) .

At continuity points, this can also be read as

P (u \leq U \leq u + d u) = f (u - c) d u + o (d u)

.

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