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    Stat Mining 22

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    Adonis Huayta
    This document discusses fundamentals of probability and random variables. It defines a random variable as a function that assigns real numbers to outcomes of an experiment. Random variables can be discrete, taking on countable values, or continuous. The probability distribution of a random variable characterizes it by giving the probabilities of all possible values. For discrete variables, this is a list of probabilities. For continuous variables, the distribution function gives the probability that the variable is less than or equal to any value, and its probability density function describes the "density mass" across possible values. Bayes' theorem allows calculating conditional probabilities from known probabilities of events and their conditions.

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    Stat Mining 22

    Uploaded by

    Adonis Huayta
    21 views1 page
    This document discusses fundamentals of probability and random variables. It defines a random variable as a function that assigns real numbers to outcomes of an experiment. Random variables can be discrete, taking on countable values, or continuous. The probability distribution of a random variable characterizes it by giving the probabilities of all possible values. For discrete variables, this is a list of probabilities. For continuous variables, the distribution function gives the probability that the variable is less than or equal to any value, and its probability density function describes the "density mass" across possible values. Bayes' theorem allows calculating conditional probabilities from known probabilities of events and their conditions.

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    This document discusses fundamentals of probability and random variables. It defines a random variable as a function that assigns real numbers to outcomes of an experiment. Random variables can be discrete, taking on countable values, or continuous. The probability distribution of a random variable characterizes it by giving the probabilities of all possible values. For discrete variables, this is a list of probabilities. For continuous variables, the distribution function gives the probability that the variable is less than or equal to any value, and its probability density function describes the "density mass" across possible values. Bayes' theorem allows calculating conditional probabilities from known probabilities of events and their conditions.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    21 views1 page

    Stat Mining 22

    Uploaded by

    Adonis Huayta
    This document discusses fundamentals of probability and random variables. It defines a random variable as a function that assigns real numbers to outcomes of an experiment. Random variables can be discrete, taking on countable values, or continuous. The probability distribution of a random variable characterizes it by giving the probabilities of all possible values. For discrete variables, this is a list of probabilities. For continuous variables, the distribution function gives the probability that the variable is less than or equal to any value, and its probability density function describes the "density mass" across possible values. Bayes' theorem allows calculating conditional probabilities from known probabilities of events and their conditions.

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    of 1

    Fundamentals 7

    The summation can be interpreted as a weighted average and consequently the marginal
    probability. Probability P(B) is sometimes called ‘average probability’ or ‘overall probability’.9
    This law usually has one common application where the events coincide with a discrete
    random variable taking each value in its range.
    Consider a set {A1, A2, …, Ak} of pairwise disjoint events whose union is the entire space.
    If P(Ai) are known and also the conditional probabilities P(B | Ai) then the conditional
    probability

    P ( B | Ai )P ( Ai )
    P ( Ai | B ) = (1.12)
    ∑
    k
    i =1
    P ( Ai )P ( B | Ai )

    This is the so-called the Bayes’ Theorem. Probability P(Ai | B) is called a posteriori whereas
    probabilities P(Ai) are called a priori.

    1.2.2 Random variables, distribution function and probability density function


    In our previous considerations, there was no specific meaning given to the event being
    observed. Actually, to every result of an experiment ξ a number will be ascribed which means
    a function is to be constructed x(ξ). Notice, that the independent variable ξ will not be a
    number but an element of set .
    A real random variable X is a function supported on the space of random events if:
    a. the set {X ≤ x} is an event for any real number x,
    b. the following equations holds:

    P{X = ∞} = P{X = −∞} = 0 (1.13)

    In other words, a measurable function assigning real numbers to every outcome of the
    experiment is called a random variable.
    Random variables will be marked in bold.
    A random variable is a discrete one if it is supported by a finite or enumerable set of numbers.
    Examples of probability distributions for discrete variables will be given in Chapter 1.2.5.
    In order to characterise a random variable, it is necessary to determine a set of its possible
    values and the corresponding probabilities.
    A function F(x), which is defined as the probability of an event {X ≤ x}, is called a
    distribution (distribution function, cumulative function) of the random variable X, i.e.

    FX(x) = P{X ≤ x} (1.14)

    The distribution is a non-decreasing monotonic function, continuous on the left and—as


    a probability—supported by a [0, 1] set.
    If a distribution FX(x) of random variable X can be defined as
    x

    FX ( x ) = ∫
    −∞
    fX u ) du
    (1.15)

    then the random variable X is continuous, its distribution is continuous and the function fX(x)
    is called a probability density function. Function fX(x) can be treated as a density mass on the

    9
    Pfeiffer (1978), Rumsey (2006).

    Book.indb 7 12/9/2013 12:21:54 PM

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