Reading Notes

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A law that we assume as true in all cases (such as Ohm's Law) and describes the "perfect" scenario. For Ohm's Law, it is almost perfectly true in small systems, but falls short in large ones or systems with noisy components.

So far we have been using input signals that we can describe mathematically. Real input signals are unpredictable and uncertain so we should treat them as random signals. Such examples are : music signals, speech, as well as any system that requires user or external input that isn't predictable. Even if they are two options, the system cannot accurately predict that an input will be one or the other.

Input signals can also be disturbed and slightly altered through their path to the system or even through it. How it is altered is random and cannot be predicted. This noise can be internal or external to the system.

System itself may have characteristics that are unknown to us that vary from time to time. An example is in a power system the load can be changed.

In large systems, components can fail which can cause the system (or part of it at least) to fail. Higher quality of components can reduce the probability that the part will fail.

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experiment -> results in an outcome. A random experiment is one where the outcome is uncertain before the experiment is performed. A single performance of an experiment is called a trial.

event -> the possible outcomes of an experiment. An equally likely event is an event that has the same probability of happening given in the same experiment. Elementary events are those whose outcomes can only be achieved one way (such as rolling one dice and getting any number 1-6). Composite events are those whose outcomes can be achieved in multiple ways (such as rolling two dice and getting their sum being odd or even).

discrete -> can be counted

continuous -> infinity many

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probability -> has too many definitions and cannot be accurately defined. Two good approaches to it though. First is relative frequency which attached some physical significance to the concept of probability but is limited because it requires the deduction of the appropriate mathematical structure for situations. Second is the axiomatic approach which treats the probability of an event as the number that certain postulates and it doesn't matter if the number can relate to anything in the real world.

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relative frequency -> linked to the frequency of occurrence of the defined event. Frequency is defined as probability.

statistical regularity -> relative frequency stabilizes for an event and approaches an unchanging number when the number of trials has no limit

The sum of the probabilities of all outcomes of an event equals 1. Each probability is between 0 and 1 where 0 represents and impossible event and 1 represents a for certain outcome.

joint probability -> the probability that two events will happen in a specific order.

Given one event A, what is the probability of a specified second event B happening.

And Also -> Event A depends on whether event B occurs

P (A|B) = ( (# of successes A) / (# of successes B) ) * ( Pr(B) )

And then -> event A doesn't not depend on event B statistically independent

P (A,B) = P(A) * P(B)

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set -> collection of objects known as elements. The largest set is called space (S). A set with no elements is known as the null set .

subset -> any set where all elements that set are contained in another set. All sets are subsets of space.

Every set is a subset of itself.

equality principle -> Set A equals set B iff every elements of A is an element of B and every element of B is an element of A.

union -> the sum of two sets consists of all elements that are elements of either one set or both sets

product of two sets -> describes the intersection of two sets where the intersection contains all elements that are shared by the two sets. If this product only contains the null space, the the two sets are mutually exclusive (disjoint) from one another.

complement -> a set that contains all the elements in space that are not in another set

DeMorgan's Laws ->

1. The complement of the product of two sets is equal to the complements of each set individually intersected together

2. The complement of the union of two sets is equal to the complements of each set individually intersected

difference of two sets -> a set that consists of the elements of set A that are not in set B. "set A without set B".

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probability space -> space where all elements are all the possible outcomes from an experiment . Subsets of this space can be identified with events (the possible outcomes of an experiments that we are looking for).

3 conditions of probability ->

1. Probability of an event is greater than or equal to 0

2. Probability of ab event in the space is equal to 1

3. If the product of two sets is null, then the probability of the union of the two sets is equal to the sum of the probabilities of each event

The product of space with null is equal to null.

The union of space with null is equal to space.

Th probability of null is 0.

The product of a set and its complement is equal to null.

The union of a set and its complement is equal to space.

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conditional probability -> the probability that an event will happen given that another event will happen. Can be found by dividing the probability of the product of the sets of outcomes and dividing by the probability that the given event happens at.

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combined experiment -> experiment that looks at two different events that do not involve the same materials such as looking at the flip of a coin and the roll of a dice at the same time. Each separate experiment has their own space. Experiments are independent of each other so the probability of an outcome of a combined experiment is the product of the probabilities of each outcome happening individually

cartesian product space -> all elements ate ordered pairs of the two spaces of a combined experiment. Can be defined as S = S1 x S2.

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bernoulli trials -> an experiment is repeated n times and we want to kind the probability of event happening k times during those n tries. To this, we multiply the binomial coefficent (factorial of number of all possibilities divided by the product of factorial of number of success and factorial of failures) and the product of the probability of failures to the power of number of failures and the probability of successes to the power of the number of successes.

DeMoivre-Laplace theorem -> used to approximate probabilities when the number of trials gets significantly large