Probability Review



1 REVIEW OF PROBABILITY 2

1.1 Some Mathematical Machinery 2

1.1.1 Set Notation 2

1.1.2 Venn Diagrams 2

1.1.3 Operations of Sets 3

1.2 Random Experiment 4

1.3 Outcomes 4

1.4 Sample Space 4

1.4.1 Examples 5

1.5 Random Event 5

1.6 Concept of Probability 5

1.7 Axioms of Probability 5

1.8 Conditional Probability 6

1.9 Law of Total Probability 7

1.10 Bayes Rule 8

1.11 Random Variable 10

1.12 Probability Function 10

1.13 Probability Mass or Density Function 12

1.14 Cumulative Probability Distribution Function (Cumulative Distribution Function) 12

1.15 Discrete Random Variables 13

1.16 Continuous Random Variables 13

1.17 Expectation 14

1.18 Jointly Distributed Random Variables 17

1.19 Some Important Discrete Distributions 18

1.19.1 Bernoulli Distribution 18

1.19.2 Binomial Distribution 18

1.19.3 Geometric Distribution 18

1.19.4 Poisson Distribution 19

1.20 Some Important Continuous Distributions 19

1.20.1 Uniform Distribution 20

1.20.2 Exponential Distribution 20

1.20.2.1 Property 1: Lack of Memory 20

1.20.2.2 Property 2: Constant Failure Rate: 22

1.20.2.3 Property 3: Combined failure rate from more than one component 22

1.20.2.4 Property 4: Relation to Poisson Distribution 23

1.20.2.5 Property 5: Detected and undetected failures. 25

1.20.3 Erlang Distribution 26

1.20.4 Normal Distribution 26

1.21 Convolution Integral 26

REVIEW OF PROBABILITY

1 Some Mathematical Machinery

Below is some useful notation and pictorial representations used in this review.

1 Set Notation

Consider the following set representation.

S={s: s is an outcome of the experiment}

Breaking down the notation:

1. { - means “the set of all elements”

2. : - means “such that”

3. text following the : - condition for inclusion of element s in the set S

Others

1. s(A – “s is an element of the set A”

2. s(A - “s is not an element of the set A”

3. A(B – “A is a subset of B”

4. [pic] - complement of the set A, i.e., [pic]

2 Venn Diagrams

Venn diagrams are pictorial representations of a set and its elements.

[pic]s((

[pic]s(A and A((

[pic][pic]

3 Operations of Sets

Unions

[pic]

[pic]

If A1, A2, …, An are events in (, then

[pic]

Intersections

[pic]

[pic]

If A1, A2, …, An are events in (, then

[pic]

2 Random Experiment

A Random Experiment is an experiment or observation which can be performed (at least in theory) any number of times under the same relevant conditions. Some examples include:

1. Toss of a fair coin

2. Toss of a fair coin twice

3. Roll of a die

4. Measured times between arrivals in an intersection

5. Count the number of rainy days in a year

3 Outcomes

An Outcome is a possible result, ( of a random experiment. For the examples listed above respectively, some example outcomes are:

1. Heads

2. Heads twice

3. Three dots on top

4. 5.3 seconds

5. 127 days

4 Sample Space

The Sample Space is the collection, (, of all possible outcomes of a random experiment. For each of the examples above, respectively, the sample space is:

1. {Heads, Tails}

2. {Heads twice, Heads then Tails, Tails then Heads, Tails Twice}

3. {One dot on top, Two dots on top, Three dots on top, Four dots on top, Five dots on top, Six dots on top}

4. {[0,(]}

5. {0, 1, 2,…365 days}

1 Examples

1. Tossing a Fair Coin - (={(: (= H or (=T} i.e., (={H,T}

2. Tossing a Fair Coin Twice - (={((1, (2): (i= H or (i =T for i=1,2} = {(T,T),(T,H),(H,T),(H,H)}

3. Time to Failure of a Transistor - (={(: 0(( ................
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