Singularly Continuous

A.3 Random Variable with Neither Density nor Probability Mass Function

Yiling Lai

2008/8/29

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Singularly Continuous

• A singularly continuous function is a continuous function that has a zero derivative almost everywhere.

• The distributions which are continuous but with no density function are said to be singular.

• An example is the Cantor function.

2

Constructing the Cantor Set 0 1

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

C2

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

C2

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

C2

C3

3

Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

C2

C3

Continue this process…

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Constructing the Cantor Set 0 1

0 3

1

3

2

1 C1

3

1

9

10 1

3

2

9

2

9

8

9

7

C2

C3

Continue this process…

1k kCC3

The length of the Cantor Set

• The first set removed is 1/3.

• The second set removed is 2/9.

• The third set removed is 4*1/27=4/27.

• The kth set removed is

• The total length removed is

• So the Cantor set, the set of points not removed, has zero “length”.

• Lebesgue measure of the Cantor set is zero.

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The Probability of Cantor Set • C1 is which has two pieces, each with

probability 1/3, and the whole set C1 has probability 2/3.

• C2 is which has four pieces, each with probability 1/9, and the whole set C1 has probability 4/9.

• At stage k, we have a set Ck that has 2k pieces, each with probability 1/3k, and the whole set Ck has probability (2/3)k.

1,

3

2

3

1,01C

1,

9

8

9

7,

3

2

3

1,

9

2

9

1,02C

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The Probability of Cantor Set • C1 is which has two pieces, each with

probability 1/3, and the whole set C1 has probability 2/3.

• C2 is which has four pieces, each with probability 1/9, and the whole set C1 has probability 4/9.

• At stage k, we have a set Ck that has 2k pieces, each with probability 1/3k, and the whole set Ck has probability (2/3)k.

1,

3

2

3

1,01C

1,

9

8

9

7,

3

2

3

1,

9

2

9

1,02C

5

03

2)()( limlim

k

kk

k

CC

Random Variable

• Fro n=1,2,…, we define

• The Probability for head on each toss is p=1/2.

• Define the random variable

1 3

2

nn

nYY

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p=1/2 p=1/2

• Case 1.1: Y1=0 with probability = 1/2

– Minimum: Y2=Y3=….=0 Y=0

– Maximum: Y2=Y3=….=1 Y=1/3

• Case 1.2: Y1=1 with probability = 1/2

– Minimum: Y2=Y3=….=0 Y=2/3

– Maximum: Y2=Y3=….=1 Y=1

...3

2

3

2

3

2

3

23

3

2

21

1

YYYYY

nn

n

0 3

1

3

21

7

• Case 2.1: Y1=0 and Y2=0 with probability = ¼

• Case 2.2: Y1=0 and Y2=1 with probability = ¼

• …….

...3

2

3

2

3

2

3

23

3

2

21

1

YYYYY

nn

n

0 3

1

9

1

9

2

4

1p

4

1p

8

• Case 2.1: Y1=0 and Y2=0 with probability = ¼

• Case 2.2: Y1=0 and Y2=1 with probability = ¼

• …….

...3

2

3

2

3

2

3

23

3

2

21

1

YYYYY

nn

n

0 3

1

9

1

9

2

4

1p

4

1p

8

• Case 2.1: Y1=0 and Y2=0 with probability = ¼

• Case 2.2: Y1=0 and Y2=1 with probability = ¼

• …….

...3

2

3

2

3

2

3

23

3

2

21

1

YYYYY

nn

n

0 3

1

9

1

9

2

4

1p

4

1p

8

• Case 2.1: Y1=0 and Y2=0 with probability = ¼

• Case 2.2: Y1=0 and Y2=1 with probability = ¼

• …….

• When we consider the first n tosses we see that the random variable Y takes values in the set Cn.

can only take values in the Cantor set .

...3

2

3

2

3

2

3

23

3

2

21

1

YYYYY

nn

n

1n nCC

0 3

1

9

1

9

2

4

1p

4

1p

8

1 3

2

nn

nYY

Does Y have a density?

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Does Y have a density?

• If Y had a density f…

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Does Y have a density?

• If Y had a density f…

• The complement of the Cantor set:

f=0

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Does Y have a density?

• If Y had a density f…

• The complement of the Cantor set:

f=0

• The Cantor set C:

C has zero Lebesgue measure, i.e. P(C)=0

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Does Y have a density?

• If Y had a density f…

• The complement of the Cantor set:

f=0

• The Cantor set C:

C has zero Lebesgue measure, i.e. P(C)=0

• So f is almost everywhere zero and

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Does Y have a density?

• If Y had a density f…

• The complement of the Cantor set:

f=0

• The Cantor set C:

C has zero Lebesgue measure, i.e. P(C)=0

• So f is almost everywhere zero and

• The function f would not integrate to one, as is required of a density.

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Does Y have a probability mass function?

• If Y had a probability mass function…

For some number we would have

• If x is not of the form

– x has a unique base-three expansion

• If x is of the form

– x has two base-three expansions.

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Does Y have a probability mass function? (cont.)

• In either case, there are at most two choices of for which

• In other words, the set has either one or two elements.

• The probability of a set with one element is zero and the probability of a set with two elements is 0+0=0.

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Does Y have a probability mass function? (cont.)

• In either case, there are at most two choices of for which

• In other words, the set has either one or two elements.

• The probability of a set with one element is zero and the probability of a set with two elements is 0+0=0.

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Does Y have a probability mass function? (cont.)

• In either case, there are at most two choices of for which

• In other words, the set has either one or two elements.

• The probability of a set with one element is zero and the probability of a set with two elements is 0+0=0.

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Y cannot have a probability mass function.

p=0

Cumulative Distribution Function

• The cumulative distribution function

satisfies

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p=1/2 p=1/2

0 3

1

3

21

0 3

1

9

1

9

2

4

1p

4

1p

Cumulative Distribution Function

• for every x, F is continuous.

•

• A non-constant continuous function whose derivative is almost everywhere zero is said to be singularly continuous.

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A Singularly Continuous Function

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A Singularly Continuous Function

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A Singularly Continuous Function

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A Singularly Continuous Function

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Singularly Continuous

• A singularly continuous function is a continuous function that has a zero derivative almost everywhere.

• An example is the Cantor function.

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