Probability Calculus Anna Janicka

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Probability Calculus Anna Janicka

lecture IV, 10.11.2020

RANDOM VARIABLES – CONT.:

CUMULATIVE DISTRIBUTION FUNCTION

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Plan for today

 Definition of the distribution of a random variable

 Description of the distribution of a random variable – examples

 Cumulative Distribution Functions

 Transformations of random variables

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Random variables – distribution

1. Definition of a random v. distribution

Different r.v. may have the same distributions

notation: X ~ 

we forget about 

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Random variables – examples

2. Examples of random variables

◼ die roll

◼ discrete distributions

◼ Binomial distribution

◼ Geometric distribution

◼ Poisson distribution

◼ uniform distribution over an interval: a continuous distribution

◼ another continuous distribution

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Continuous random variables

3. Definition of a continuous random variable and a density function

4. The properties of density functions

◼ nonnegative

◼ normalized

◼ determines the distribution unequivocally

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Random variable examples – cont.

5. More examples of continuous random variables

◼ uniform distribution

◼ exponential distribution

◼ standard normal distribution

◼ general normal distribution

◼ (Dirac delta)

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Random variables – the CDF

1. The definition of a CDF

depends on the distribution only!

→ CDF of distribution

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Random variables – the CDF

2. Examples of CDFs

◼ Dirac delta

◼ Two-point distribution – discrete distribution

◼ Exponential distribution

◼ Normal distribution – no simple form…

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CDFs

3. Properties of the CDF

4. CDF → distribution

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CDFs – cont.

5. A CDF of a discrete distribution 6. Further properties of the CDF:

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CDFs – cont (2)

7. CDF → density

8. Examples

◼ uniform distribution

◼ distribution that is neither discrete nor continuous

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Transformation of random variables

9. Well-behaved transformations of continuous variables

10. Example

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