Probability Theory on Coin Toss Space: Finite Spaces, Random Variables, and Expectations, Study notes of Probability and Statistics

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Probability Theory on Coin Toss Space

1 Finite Probability Spaces

2 Random Variables, Distributions, and Expectations

3 Conditional Expectations

Probability Theory on Coin Toss Space

1 Finite Probability Spaces

2 Random Variables, Distributions, and Expectations

3 Conditional Expectations

Inspiration

• A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes
• Let us discuss the binomial model we have studied so far through a very simple example
• Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as

Ω = {HHH, HHT , HTH, THH, HTT , THT , TTH, TTT }

• (^) Assume that the probability of a head is p and the probability of a tail is q = 1 − p
• (^) Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are

P[HHH] = p^3 , P[HHT ] = P[HTH] = P[THH] = p^2 q, P[TTT ] = q^3 , P[HTT ] = P[THT ] = P[TTH] = pq^2

Inspiration

• A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes
• Let us discuss the binomial model we have studied so far through a very simple example
• Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as

Ω = {HHH, HHT , HTH, THH, HTT , THT , TTH, TTT }

• (^) Assume that the probability of a head is p and the probability of a tail is q = 1 − p
• (^) Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are

P[HHH] = p^3 , P[HHT ] = P[HTH] = P[THH] = p^2 q, P[TTT ] = q^3 , P[HTT ] = P[THT ] = P[TTH] = pq^2

Inspiration

• A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes
• Let us discuss the binomial model we have studied so far through a very simple example
• Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as

Ω = {HHH, HHT , HTH, THH, HTT , THT , TTH, TTT }

• (^) Assume that the probability of a head is p and the probability of a tail is q = 1 − p
• (^) Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are

P[HHH] = p^3 , P[HHT ] = P[HTH] = P[THH] = p^2 q, P[TTT ] = q^3 , P[HTT ] = P[THT ] = P[TTH] = pq^2

An Example (cont’d)

• (^) The subsets of Ω are called events, e.g.,

”The first toss is a head” = {ω ∈ Ω : ω 1 = H} = {HHH, HTH, HTT }

• The probability of an event is then

P[”The first toss is a head”] = P[HHH] + P[HTH] + P[HTT ] = p

• The final answer agrees with our intuition - which is good

An Example (cont’d)

• (^) The subsets of Ω are called events, e.g.,

”The first toss is a head” = {ω ∈ Ω : ω 1 = H} = {HHH, HTH, HTT }

• The probability of an event is then

P[”The first toss is a head”] = P[HHH] + P[HTH] + P[HTT ] = p

• The final answer agrees with our intuition - which is good

Definitions

• A finite probability space consists of a sample space Ω and a probability measure P. The sample space Ω is a nonempty finite set and the probability measure P is a function which assigns to each element ω in Ω a number in [0, 1] so that ∑

ω∈Ω

P[ω] = 1.

An event is a subset of Ω. We define the probability of an event A as

P[A] =

ω∈A

P[ω]

• Note:

P[Ω] = 1

and if A ∩ B = ∅ P[A ∪ B] = P[A] + P[B]

Definitions

• A finite probability space consists of a sample space Ω and a probability measure P. The sample space Ω is a nonempty finite set and the probability measure P is a function which assigns to each element ω in Ω a number in [0, 1] so that ∑

ω∈Ω

P[ω] = 1.

An event is a subset of Ω. We define the probability of an event A as

P[A] =

ω∈A

P[ω]

• Note:

P[Ω] = 1

and if A ∩ B = ∅ P[A ∪ B] = P[A] + P[B]

Probability Theory on Coin Toss Space

1 Finite Probability Spaces

2 Random Variables, Distributions, and Expectations

3 Conditional Expectations

Random variables

• (^) Definition. A random variable is a real-valued function defined on Ω
• (^) Example (Stock prices) Let the sample space Ω be the one corresponding to the three coin tosses. We define the stock prices on days 0, 1 , 2 as follows:

S 0 (ω 1 ω 2 ω 3 ) = 4 for all ω 1 ω 2 ω 3 ∈ Ω

S 1 (ω 1 ω 2 ω 3 ) =

8 for ω 1 = H 2 for ω 1 = T

S 2 (ω 1 ω 2 ω 3 ) =

16 for ω 1 = ω 2 = H 4 for ω 1 6 = ω 2 1 for ω 1 = ω 2 = H

Distributions

• The distribution of a random variable is a specification of the probabilities that the random variable takes various values.
• Following up on the previous example, we have

P[S 2 = 16] = P{ω ∈ Ω : S 2 (ω) = 16} = P{ω = ω 1 ω 2 ω 3 ∈ Ω : ω 1 = ω 2 } = P[HHH] + P[HHT ] = p^2

• Is is customary to write the distribution of a random variable on a finite probability space as a table of probabilities that the random variable takes various values.

Distributions

• The distribution of a random variable is a specification of the probabilities that the random variable takes various values.
• Following up on the previous example, we have

P[S 2 = 16] = P{ω ∈ Ω : S 2 (ω) = 16} = P{ω = ω 1 ω 2 ω 3 ∈ Ω : ω 1 = ω 2 } = P[HHH] + P[HHT ] = p^2

• Is is customary to write the distribution of a random variable on a finite probability space as a table of probabilities that the random variable takes various values.

Expectations

• (^) Let a random variable X be defined on a finite probability space (Ω, P). The expectation (or expected value) of X is defined as

E[X ] =

ω∈Ω

X (ω)P[ω]

• (^) The variance of X is

Var [X ] = E[(X − E[X ])^2 ]

• (^) Note: The expectation is linear, i.e., if X and Y are random variables on the same probability space and c and d are constants, then

E[cX + dY ] = cE[X ] + dE[Y ]