Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 9 / a

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

i

Solution

For a continuous random variable with probability density function

f_{X}

, its mean is

μ = E [X] = \int_{- \infty}^{\infty} x f_{X} (x) d x,

provided

\int_{- \infty}^{\infty} ∣ x ∣ f_{X} (x) d x < \infty

. Its variance is the expectation of the squared deviation from its mean:

var (X) = E [(X - μ)^{2}] = \int_{- \infty}^{\infty} (x - μ)^{2} f_{X} (x) d x .

This is finite when

E [X^{2}] < \infty

; otherwise it may be

+ \infty

if the finite mean exists.

ii

Solution

Assume a finite second moment, so the following expectations are finite. Expand the square and use linearity of expectation:

var (X) = E [X^{2} - 2 μ X + μ^{2}] = E [X^{2}] - 2 μ E [X] + μ^{2} = E [X^{2}] - μ^{2} .

Thus

var (X) = E [X^{2}] - (E [X])^{2} .

The same equality holds in the extended sense when the mean is finite and the second moment is infinite.

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