Understanding Random Variables: Definitions, Properties, and Distributions, Study notes of Introduction to Econometrics

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Random Variables

Helle Bunzel

Overview

? We will go over most of the general concepts.

? Our coverage will be mostly without proofs.

? Outline:

1. Review of univariate random variables, their distributions and the properties of those distributions.
2. Review of multidimensional random variables and their properties.
3. Finally we'll look at distributions of quadratic forms.

Random Variables

Denition 4 If X : S! R is a real valued function having the elements

of S as its domain, then X is called a random variable.

? There are two main types of random variables:

 A random variable is discrete if the set of outcomes is either nite

or countably nite.

 A random variable is continuous if the set of outcomes is innitely

divisible.

? Finally, PX is a probability function dened such that it assigns a prob-

ability to all possible events.

? Probabilities can be assigned to different outcomes.

? Data: Outcome of a random variable.

? P (X = x) is the probability that the random variable X takes on the

value x.

Density Fuctions

? Dene a function f (x) from S to R that satises:

1. For discrete distributions, 0  f (x)  1,

and for continuous 0  f (x)

2. For discrete distributions ∑x f (x) = 1,

and for continuous

R

f (x) dx = 1

? For a discrete random variable, f (x) = P (X = x).

 Note that if x is not a possible outcome (not in S), then f (x) = 0.

? For continuous distributions R P (X = x) = 0, but P (A  X  B) =

B

A f^ (x)^ dx

 NOTE: f (x) CANNOT be interpreted as point probability.

Example

Theorem 5 Fundamental Theorem of Calculus: ∂ ∂ b

Z (^) b a

f (x) dx = f (b)

∂ a

Z (^) b

a

f (x) dx = f (a)

? Use this:

∂ b

Z (^) b a

f (x) dx = ∂

∂ b

b a

f (b) = 0.

? This implies that

f (x) =

0.1, x 2 [0; 10]

0 otherwise =^ 0.1^
^1 fx^2 [0,10]g

? What is the probability that an accident occurs on the rst half of the

stretch of highway?

Example

? What is the probability of A = [1, 2] [ [7, 9]?

? What is the probability of B = [0, 2) [ (7, 8) [ [9, 10]?

? Note: Probability 0 vs. Impossibility.

Cumulative Distribution Functions

? The cumulative probability function F (x) is dened as P (X  x)

 For the discrete this is F (x) = ∑yx f (y) = ∑yx P (X = y)

 For the continuous it is F (x) =

R x

∞ f^ (y)^ dy

 Note that this denition implies that f (x) = ∂∂ x F (x)

? The properties of F are as follows:

1. 0  F (x)  1

2. if x  y then F (x)  F (y)

3. F (∞) = 1

4. F (∞) = 0

5. P (a  X  b) = F (b) F (a)

? Drawings.

Multivariate Random Variables

? A multivariate random variable, X can be written as

X =

X 1

X 2

Xn

where Xi, i = 1, ..., n are random variables.

? Examples of multivariate random variables?

 Multiple characteristics.

 Cars

 People

? The density of X is denoted by f (x 1 , x 2 , .., xn)

? Discrete: If all the individual r.v.s are discrete.

? Continuous: If all the individual r.v.s are continuous.

Example

? A pair of dice, two colors. Red, green. X 1 : Number of eyes on red.

X 2 : Sum of eyes on red and green.

? Elements in S:

? What is the density of this stochastic variable?

? What is the probability that red is two or less and the sum is 5 or less?

P (A) =

2

x 1 = 1

5

x 2 =x 1 + 1

f (x 1 , x 2 ) = 7

Example

? Big screen TV production.

? TVs 3  4 feet.

? Coating machine produces a aw. Each point on the screen is equally

likely to recieve the aw.

? Label the area [1.5, 1.5]  [2, 2].

? What is the density of aws?

? What is the probability that the aw ends in an area with width W and

height H?

? We are looking for the function where

Z (^) d c

Z (^) b a

f (x, y) dxdy =

(b a) (d c)

for all 2  a < b  2 and 1.5  c < d  1.5.

Multivariate CDFs

? When X is discrete, the multivariate CDF is denied as

F (b 1 , ..., bn) = P (Xi  bi, i = 1, .., n) = ∑

x 1 b 1

xnbn

f (x 1 , ..., xn)

? When X is continuous, the multivariate CDF is denied as

F (b 1 , ..., bn) = P (Xi  bi, i = 1, .., n) =

Z (^) bn ∞

Z (^) b 1 ∞

f (x 1 , ..., xn)

? Properties of the multivariate CDF:

1. limbi!∞F (b 1 , ..., bn) = 0, i = 1, ..., n.

2. limbi!∞,i=1,...,n F (b 1 , ..., bn) = 1.

3. F (a) < F (b) for a < b.

Multivariate CDFs

? In the continuous case, we can move from CDF to pdf much as before:

Theorem 6 f (x 1 , ..., xn) =

 (^) ∂ n

∂ x 1 ... ∂ xn F^ (x^1 , ...,^ xn)^ where^ f^ is continuous

0 elsewhere

? In the discrete case it is much more complicated. Two dimensional

example:

 (X, Y) : binary discrete variable with joint cumulative distribution

function F(X, Y)

 x 1 < x 2 < .. and y 1 < y 2 < ...possible outcomes of X and Y

 f (x 1 , y 1 ) = F (x 1 , y 1 )

 f

x 1 , yj

= F

x 1 , yj

F

x 1 , yj 1

, j  2.

 f (xi, y 1 ) = F (xi, y 1 ) F (xi 1 , y 1 ) , i  2.

 f

xi, yj

= F

xi, yj

F

xi, yj 1

F

xi 1 , yj

+ F

xi 1 , yj 1

i, j  2

Example

Z (^) b 2

∞

1 [1.5,1.5] (x 2 ) dx 2 = 1 [1.5,∞) (b 2 )

Z (^) 1.

1.

1 dx 2 + 1 [1.5,1.5) (b 2 )

Z (^) b 2

1.

1 dx 2

= 3
1 [1.5,∞) (b 2 ) + (b 2 + 1.5) 1 [1.5,1.5) (b 2 )

F (b 1 , b 2 ) =

Z (^) b 1

∞

1 [2,2] (x 1 )

Z (^) b 2

∞

1 [1.5,1.5] (x 2 ) dx 2 dx 1

1 [1.5,∞) (b 2 )

Z (^) b 1

∞

1 [2,2] (x 1 ) dx 1

+b^2 +^ 1.

1 [1.5,1.5) (b 2 )

Z (^) b 1

∞

1 [2,2] (x 1 ) dx 1

Z (^) b 1

∞

1 [2,2] (x 1 ) dx 1 = 1 [2,2] (b 1 )

Z (^) b 1

2

1 dx 1 + 1 [2,∞] (b 1 )

Z (^2)

2

1 dx 1

= 1 [2,2] (b 1 ) (b 1 + 2 ) + 4
1 [2,∞] (b 1 )

Example

F (b 1 , b 2 ) = 1

1 [1.5,∞) (b 2 )

h

1 [2,2] (b 1 ) (b 1 + 2 ) + 4
1 [2,∞] (b 1 )

i

+b^2 +^ 1.

1 [1.5,1.5) (b 2 )

h

1 [2,2] (b 1 ) (b 1 + 2 ) + 4
1 [2,∞] (b 1 )

i

b 1 + 2 )

4 1 [1.5,∞)^ (b^2 )^1 [2,2]^ (b^1 ) +^1 [1.5,∞)^ (b^2 )^1 [2,∞]^ (b^1 )

+(b^2 +^ 1.5) (b^1 +^2 )

1 [1.5,1.5) (b 2 ) 1 [2,2] (b 1 )

+b^2 +^ 1.

1 [1.5,1.5) (b 2 ) 1 [2,∞] (b 1 )