Dr Mike

Puddephat

Online

Taken from Mike Puddephat's PhD, in this article fundamental results regarding the expectation and variance of random variables (discrete or continuous) are stated and proved.

This article follows on from An Introduction to Random Variables (Part 1). Here, fundamental results regarding the expectation and variance of random variables (discrete or continuous) are stated and proved.

**Property 1**

For some constant *c* ∈ ℜ and random variable *X*,

**E**(*cX*) = *c***E**(*X*). (1)

*Proof:*

From Part 1-(1), for *X* discrete

while from Part 1-(8), for *X* continuous

**Property 2**

For discrete or continuous random variables *X* and *Y*,

**E**(*X* + *Y*) = **E**(*X*) + **E**(*Y*). (2)

*Proof:*

Suppose *X* and *Y* have joint mass function *f _{X,Y}* : ℜ

Noting Part 1-(8), for *X* and *Y* continuous the proof begins

where *f _{X,Y}*(

**Property 3**

For discrete or continuous independent random variables *X* and *Y*,

**E**(*XY*) = **E**(*X*)⋅**E**(*Y*) (3)

*Proof:*

The proof of (3) is first presented for discrete random variables *X* and *Y*. Let *X* and *Y* have joint mass function *f _{X,Y}*(

**P**(*X* = *x* and *Y* = *y*) = **P**((*X* = *x*) ∩ (*Y* = *y*)) = **P**(*X* = *x*)⋅**P**(*Y* = *y*),

so that *f _{X,Y}*(

For *X* and *Y* continuous, the proof begins

where *f _{X}*

**Property 4**

For the discrete or continuous random variable *X*,

(4)

*Proof:*

The proof of (4) holds for *X* discrete or continuous. As a shorthand notation, let μ = **E**(*X*). Then,

**Property 5**

For the discrete or continuous random variable *X* and the constant c ∈ ℜ,

Var(*cX*) = *c*^{2}⋅Var(*X*). (5)

*Proof:*

The proof of (5) holds for *X* discrete or continuous. Again, let μ = **E**(*X*). Then,

Var(*cX*) = **E**((*cX* - *c**μ*)^{2}) = **E**(*c*^{2}⋅(*X* - *μ*)^{2}) = *c*^{2}⋅**E**((*X* - *μ*)^{2}) = *c*^{2}⋅Var(*X*).

**Property 6**

For discrete or continuous independent random variables *X* and *Y*,

Var(*X* + *Y*) = Var(*X*) + Var(*Y*).(6)

*Proof:*

The proof of (6) holds for *X* and *Y* discrete or continuous. As a shorthand notation, let *μ*_{X } = **E**(*X*) and *μ _{Y}* =

However, from (3), if *X* and *Y* are independent random variables, then

- Posted: 14 February, 2012

- Posted: 1 December, 2011

- Posted: 29 June, 2010

- Posted: 24 June, 2010

- Posted: 23 June, 2010

- Posted: 21 June, 2010

- Posted: 17 June, 2010

- Posted: 17 June, 2010

- Posted: 15 June, 2010

- Posted: 14 June, 2010