Probability theory
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Probability Theory. Part 2: Random Variables. Random Variables. The Notion of a Random Variable The outcome is not always a number Assign a numerical value to the outcome of the experiment Definition

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Probability Theory

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Probability theory

Probability Theory

Part 2: Random Variables


Random variables

Random Variables

  • The Notion of a Random Variable

    • The outcome is not always a number

    • Assign a numerical value to the outcome of the experiment

  • Definition

    • A function X which assigns a real number X(ζ) to each outcome ζ in the sample space of a random experiment

S

ζ

X(ζ) = x

x

Sx


Cumulative distribution function

Cumulative Distribution Function

  • Defined as the probability of the event {X≤x}

  • Properties

Fx(x)

1

x

Fx(x)

1

¾

½

¼

3

x

0

1

2


Types of random variables

Continuous

Probability Density Function

Discrete

Probability Mass Function

Types of Random Variables


Probability density function

Probability Density Function

  • The pdf is computed from

  • Properties

  • For discrete r.v.

fX(x)

fX(x)

dx

x


Conditional distribution

The conditional distribution function of X given the event B

The conditional pdf is

The distribution function can be written as a weighted sum of conditional distribution functions

where Ai mutally exclusive and exhaustive events

Conditional Distribution


Expected value and variance

The expected value or mean of X is

Properties

The variance of X is

The standard deviation of X is

Properties

Expected Value and Variance


More on mean and variance

Physical Meaning

If pmf is a set of point masses, then the expected value μ is the center of mass, and the standard deviation σ is a measure of how far values of x are likely to depart from μ

Markov’s Inequality

Chebyshev’s Inequality

Both provide crude upper bounds for certain r.v.’s but might be useful when little is known for the r.v.

More on Mean and Variance


Joint distributions

Joint Probability Mass Function of X, Y

Probability of event A

Marginal PMFs (events involving each rv in isolation)

Joint CMF of X, Y

Marginal CMFs

Joint Distributions


Conditional probability and expectation

The conditional CDF of Y given the event {X=x} is

The conditional PDF of Y given the event {X=x} is

The conditional expectation of Y given X=x is

Conditional Probability and Expectation


Independence of two random variables

X and Y are independent if {X ≤ x} and {Y ≤ y} are independent for every combination of x, y

Conditional Probability of independent R.V.s

Independence of two Random Variables


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