ECE 313

奇怪的ECE 313，他为什么属于ECE而不是属于MATH呢

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probability space

It’s a triplet

• : A nonempty set, each element of is called an outcome and is called the sample space. The number of is called the cardinality of
• : read as Script F, a set of all subsets of , also call it events.
• : a probability measure on F. is the probability of event ().

We use or to mean the complement of A.

Event axioms:

• is an event ().
• If is an event then is an event ().
• If and are events then is an event ().

Probability axioms:

• .
• if and and are mutually exclusive , then .

Calculate the size of various sets

Principle of counting: If there are ways to select one variable and ways to select another varaible, and if these two selections can be made independently, then there is a total of ways to make the pair of selections.

n choose k:

A random variable is a real-valued function on :

• pmf (probability mass function):

The mean of a random variable:
The mean (also called expectation) of a random variable with pmf is denoted by and is defined by , where is the list of possible values of .
The general formula for the mean of a function, , of , is .
.
The variance and standard deviation of a random variable:
The variance of a random variable is a measure of how spread out the pmf of is. Letting , the variance is defined by:
,
The standardized version of is the random variable , and

Conditional probabilities

The conditional probability of B given A is defined by:

Mutually independent events

Event is independent of event if .

Events , and are pairwise independent if

Events , and are independent if ther are pairwise independent and if

Discrete-type indepent random variables

Random variables and are independent if any event of the form is independent of any event of the form . ()

Binomial distribution

A random variable is said to have the Bernoulli distribution with parameter , where , if and . ,

Suppose independent Bernoulli trials are conducted, each resulting in a one with probability and a zero with probability . Let denote the total number of ones occurring in the trials. The pmf of is
and

Geometric distribution

Do Bernoulli trials untill the outcome of a trial is one. L denote the number of trials conducted. The pmf of L is: and .
,

Negative binomial distribution

Let denotes the number of trials required for ones, and the last trail must be one. Let , and let . The event is determined by the outcomes of the ffirst n trials. The event is true iff there are ones and zeros in the first trials, and trail is one. Therefore, the pmf of is given by
,

Poisson disttribution

The Poission probability distribution with parameter is the one with pmf . It’s a good approximation for a binomial distribution with parameters and , when is very large, is very small, and .

Examples:

• Radio active emissions in a fixed time interval: n is the number of uranium atoms in a rock sample, and p is the probability that any particular one of those atoms emits a particle in a one minute period.
• Incoming phone calls in a fixed time interval: n is the number of people with cell phones within the access region of one base station, and p is the probability that a given such person will make a call within the next minute.
• Misspelled words in a document: n is the number of words in a document and p is the probability that a given word is misspelled.

Maximum likelihood parameter estimation

For a random variable , and that the pmf of is , where is a parameter. The probability of k being the observed value for X. The likelihood is . The maximum likelihood estimate of for observation , denoted by , is the value of that maximizes the likelihood, , with respect to . (Give k, find (or in some Bernoulli trials) to make biggest, the value of ).

Markov and Chebychev inequalities and confidence intervals

Markov’s inequality:
Chebychev inequality:

The law of total probability