abdellah Chkifa
Some of my presentations
Probability Theory on Coin Toss Space
Outline
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Finite probability spaces
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Random variables, distribution, expectation
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Conditional Expectations
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Martingales
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Markov Processes
Finite Probability Spaces
Definition
A finite probability space is used to model a situation in which a random experiment with finitely many possible outcomes is conducted. We tossed a coin a finite number of times. For example, we toss the coin twice, the set of all possible outcomes is
\Omega = \{HH,HT,TH,TT\}
Ω={HH,HT,TH,TT}
Suppose that on each toss the probability of a head is p and the probability of a tail is q = 1 - p. We assume the tosses are independent, and so the probabilities of the individual elements. The subsets of fi are called events. For example, the event "The first toss is a head" is .The probability of an event by summing the probabilities of the elements in the event, i.e.,
{\mathbb P}(\text{First toss is a head})=
{\mathbb P}(HT)+
{\mathbb P}(HH)
P(First toss is a head)=P(HT)+P(HH)
Definition
Definition: A finite probability space consists of a sample space and a probability measure . The sample space is a nonempty finite set and the probability measure is a function that assigns to each element a number in [0,1] so that
An event $A$ is a subset of $\Omega$ and the probability of $A$ is defined by
\mathbb P
P
\displaystyle\sum_{\omega \in \Omega} {\mathbb P} (\omega)=1
ω∈Ω∑P(ω)=1
{\mathbb P}(A) =\displaystyle\sum_{\omega \in A} {\mathbb P} (\omega)
P(A)=ω∈A∑P(ω)
Random Variables, Distributions, and Expectations
Definition Let $(\Omega,{\mathbb P})$ be a finite probability space. A random variable is a real-valued function defined on $\Omega$.
Definition Let $X$ be a random variable defined on a finite probability space $(\Omega,{\mathbb P})$. The expectation (or expected valued of X) is defined to by
{\mathbb E}[X] = \sum_{\omega\in\Omega} X_\omega {\mathbb P}(w)
E[X]=∑ω∈ΩXωP(w)
The variance of $X$ is defined by
\text{Var}[X] = {\mathbb E}[(X-{\mathbb E}[X])^2]
Var[X]=E[(X−E[X])2]
Properties: The expectation and variance satisfies
{\mathbb E}[c_1 X+ c_2Y] = c_1{\mathbb E}[X]+
c_2{\mathbb E}[Y]
E[c1X+c2Y]=c1E[X]+c2E[Y]
\text{Var}[ X] =
{\mathbb E}[X^2]-
{\mathbb E}[X]^2
Var[X]=E[X2]−E[X]2
Theorem (Jensen inequality): Let $X$ be a random variable on a finite probability space and $\phi$ a convex function. Then
{\mathbb E}[\phi(X)]\geq \phi({\mathbb E}[X])
E[ϕ(X)]≥ϕ(E[X])
Probability Theory on Coin Toss Space
Current deck
By abdellah Chkifa
Current deck
- 812
