1) Random Variables Flashcards

1
Q

What is a sample space

A

The set of possible outcomes , Ω

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2
Q

What is an event

A

A susbet of the sample space

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3
Q

Describe some of the common notation used in probability

A
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4
Q

What does it mean if A and B are disjoint

A

A ∩ B = ∅

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5
Q

What is the Probability Space

A

Conatins 3 elements (Ω, F, P) -
* Ω - sample space
* F - subsets of Ω
* P - probability measure

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6
Q

When is a collection F of subsets of Ω a σ-algebra

A
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7
Q

What is a probability measure P on (Ω, F)

A

A function P: F → [0, 1] satisfying:
(i) P(Ω) = 1,
(ii) P(∅) = 0

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8
Q

When is a function X : Ω → R measurable

A

If the sets { ω | X(ω) ⩽ x } ∈ F for all x ∈ R

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9
Q

What is a random variable

A

A measurable function X : Ω → R

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10
Q

What is the range of X

A

The set X(Ω) of all possible values of X

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11
Q

What is the cumulative distribution function (cdf)

A
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12
Q

What are the properties of the probability of a cdf

A
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13
Q

What are the properties of the cdf

A
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14
Q

What is a discrete random variable

A

If the range of the random variable X can only assume countably many values, RX = {x1, x2, … }

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15
Q

What is the probability mass function (pmf)

A
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16
Q

What are the properties of the pmf

17
Q

What is the Bernoulli Distribution

18
Q

What is the Binomial Distribution

19
Q

What is the Geometric Distribution

20
Q

What is the Poisson Distribution

21
Q

What is the uniform distribution (discrete)

22
Q

What is the cdf of discrete random variables

A

The c.d.f. of a discrete random variable is a step function

23
Q

What is the probability density function (pdf)

A

The function fX : R → [0, ∞) such that

24
Q

What is the relationship between pdf and cdf

25
What are the properties of the pdf
26
What is the uniform distribution (continuous)
27
What is the exponential distribution
28
What is the normal distribution
29
What is the pdf of an induced random variable Y = g(X)
Suppose that g is either strictly increasing or decreasing
30
Describe the proof of the pdf of an induced random variable Y = g(X)