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Probability Theory

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Probability Theory

Part 2: 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

- 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

Continuous

Probability Density Function

Discrete

Probability Mass Function

- The pdf is computed from
- Properties
- For discrete r.v.

fX(x)

fX(x)

dx

x

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

The expected value or mean of X is

Properties

The variance of X is

The standard deviation of X is

Properties

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.

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

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

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