Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 18 / b

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

i

Solution

Interpreting

f

as the probability density function of the random variable, for

α \leq β

one has

P (α \leq X \leq β) = \int_{α}^{β} f (x) d x .

A single point has zero probability, so including or excluding either endpoint does not change this result.

ii

Solution

When the relevant moments are finite, the mean and variance are

μ = E [X] = \int_{- \infty}^{\infty} x f (x) d x, σ^{2} = E [(X - μ)^{2}] = \int_{- \infty}^{\infty} (x - μ)^{2} f (x) d x .

Equivalently,

σ^{2} = \int x^{2} f (x) d x - μ^{2}

. The mean requires an integrable absolute first moment; finite second moment guarantees both quantities are finite.

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