Understanding Random Variables and Distribution Functions, Study notes of Statistics

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Topic 7

Random Variables and Distribution Functions

Distribution Functions

Outline

Definition of a Random Variable

Distribution Functions

Discrete Random Variables

Continuous Random Variables

Properties of Distribution Functions

Definition of a Random Variable

A random variable is a real valued function from the probability space.

X : Ω → R.

Typically, we shall use capital letters near the end of the alphabet, e.g., X , Y , Z for random variables. The range of a random variable is called the state space. Exercise. Give some random variables on the following probability spaces, Ω.

1. Roll a die 3 times and consider the sample space Ω = {(i, j, k); i, j, k = 1, 2 , 3 , 4 , 5 , 6 }.
2. Flip a coin 10 times and consider the sample space Ω, the set of 10-tuples of heads and tails. We can create new random variables via composition of functions:

ω 7 → X (ω) 7 → g (X (ω))

Thus, if X is a random variable, then so are

X 2 , exp αX ,

X 2 + 1, tan^2 X , bX c (^) 4 / 11

Distribution Functions

A (cumulative) distribution function of a random variable X is defined by

FX (x) = P{ω ∈ Ω; X (ω) ≤ x} = P{X ≤ x}.

For the complement of {X ≤ x}, we have the survival function

F (^) X (x) = P{X > x} = 1 − P{X ≤ x} = 1 − FX (x).

Choose a < b, then the event {X ≤ a} ⊂ {X ≤ b}. Their set theoretic difference

{X ≤ b} \ {X ≤ a} = {a < X ≤ b}.

Consequently, by the difference rule for probabilities,

P{a < X ≤ b} = P({X ≤ b} \ {X ≤ a}) = P{X ≤ b} − P{X ≤ a} = FX (b) − FX (a).

In particular, FX is non-decreasing.

Distribution Functions

Notice that the distribution function

• (^) is constant in between the possible values for X ,
• has a jump size at x is equal to P{X = x}, and
• (^) is right continuous.

Call X a discrete random variable if its distribution function FX has these properties. Examples. 3 36

= P{X = 4} = FX (4) − FX (4−) = 6

P{ 4 < X ≤ 7 } = FX (7) − FX (4) =

36 −^

36 =^

P{ 4 ≤ X ≤ 7 } = FX (7) − FX (4−) =^21

=^18

=^1

Distribution Functions

Exercise.

1. Flip a fair coins 3 times. Let X be the number of heads. Under equally likely outcomes, find P{X = x} for x = 0, 1 , 2 , and 3. and use this to sketch a graph of the distribution function FX.
2. Deal 5 cards out of a deck of 52. Let X be the number of ♦. Under equally likely outcomes, use the choose function in R to determine

P{X = x} for x = 0, 1 , 2 , 3 , 4 , and 5.

and use this to sketch a graph of the distribution function FX.

Distribution Functions

Exercise.

1. Find the probability that the dart no more than 1/2 unit from the center.
2. Find the probability that the dart lands further 1/3 unit but no more than 2/ unit from the center.
3. Find the median, x 1 / 2 so that P{X ≤ x 1 / 2 } = 1/2.

Definition. X is continuous random variable if it has a cumulative distribution function FX that is differentiable.

Properties of Distribution Functions

A distribution function FX has the property that it is right continuous, starts at 0, ends at 1, and does not decrease with increasing values of x.

In mathematical terms,

• (^) For every a, limx→a+ FX (x) = FX (a).
• (^) limx→−∞ FX (x) = 0.
• limx→∞ FX (x) = 1.
• (^) For every a, b satisfying a < b,

FX (a) ≤ Fx (b). 0 5 10 15 20 25

0.^ 0.^ 0.^ 0.^ 0.^

x

distribution function

0 5 10 15 20 25

0.^ 0.^ 0.^ 0.^ 0.^

0 5 10 15 20 25

0.^ 0.^ 0.^ 0.^ 0.^