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    Probability Distribution

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    04 Shaina Sachdeva
    The document discusses different probability distributions including binomial, Poisson, and normal distributions. It provides: 1) Definitions and examples of each distribution, noting that the binomial models outcomes of success/failure experiments, Poisson models rare events, and normal is the most common distribution. 2) Key properties and functions of each, such as the binomial having mean np and variance npq, and Poisson and normal distributions both having mean equal to variance. 3) Common applications of each, like the binomial for coin tosses and tests, Poisson for rare events like accidents, and normal distribution modeling many physical and social phenomena.

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    Probability Distribution

    Uploaded by

    04 Shaina Sachdeva
    44 views15 pages
    The document discusses different probability distributions including binomial, Poisson, and normal distributions. It provides: 1) Definitions and examples of each distribution, noting that the binomial models outcomes of success/failure experiments, Poisson models rare events, and normal is the most common distribution. 2) Key properties and functions of each, such as the binomial having mean np and variance npq, and Poisson and normal distributions both having mean equal to variance. 3) Common applications of each, like the binomial for coin tosses and tests, Poisson for rare events like accidents, and normal distribution modeling many physical and social phenomena.

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    PROBABILITY DISTRIBUTION

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    The document discusses different probability distributions including binomial, Poisson, and normal distributions. It provides: 1) Definitions and examples of each distribution, noting that the binomial models outcomes of success/failure experiments, Poisson models rare events, and normal is the most common distribution. 2) Key properties and functions of each, such as the binomial having mean np and variance npq, and Poisson and normal distributions both having mean equal to variance. 3) Common applications of each, like the binomial for coin tosses and tests, Poisson for rare events like accidents, and normal distribution modeling many physical and social phenomena.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    44 views15 pages

    Probability Distribution

    Uploaded by

    04 Shaina Sachdeva
    The document discusses different probability distributions including binomial, Poisson, and normal distributions. It provides: 1) Definitions and examples of each distribution, noting that the binomial models outcomes of success/failure experiments, Poisson models rare events, and normal is the most common distribution. 2) Key properties and functions of each, such as the binomial having mean np and variance npq, and Poisson and normal distributions both having mean equal to variance. 3) Common applications of each, like the binomial for coin tosses and tests, Poisson for rare events like accidents, and normal distribution modeling many physical and social phenomena.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 15

    PROBABILITY DISTRIBUTION

    • Probability Distribution is obtained in


    relation to a random variable by some
    mathematical model.
    • We can find the expected frequencies by
    applying the models.
    • Also called, Theoretical or expected
    frequency distributions.
    • Examples include- Binomial , Poisson
    and Normal distribution.
    BINOMIAL DISTRIBUTION
    A binomial distribution can be thought of as simply
    the probability of a SUCCESS or FAILURE outcome
    in an experiment or survey that is repeated multiple
    times. The binomial is a type of distribution that
    has two possible outcomes (the prefix “bi” means two,
    or twice). For example, a coin toss has only two
    possible outcomes: heads or tails and taking a test
    could have two possible outcomes: pass or fail.
    This model is based on the binomial theorem and was
    propounded by Jacob Bernouli in 1700.
    APPLICATIONS OF BINOMIAL
    DISTRIBUTION-
    1) Where no. of trials are too large but finite.
    2) Every trial is independent.
    3) Events are mutually exclusive and
    exhaustive.
    4) Every trial has only two outcomes .
    Occurrence of one will be known as
    success indicated by ‘p’ and occurrence
    of another is known as failure indicated
    by ‘q’.
    PROPERTIES OF BINOMIAL
    DISTRIBUTION-
    1) Mean= np
    2) Variance = npq
    3) Standard deviation = √npq
    4) Variance is always less than mean.
    5) Variance will be highest when p=q
    6) Mode= (n+1)p
    i) Integer- Bi-modal (2 modes)- a) (n+1)p
    & b)(n+1)p-1
    ii) Non-integer- Uni- Modal( 1 mode)- Greatest
    Integer in (n+1)p
    FUNCTION OF BINOMIAL
    DISTRIBUTION-
    P(x) = nCr · pr (1 − p)n−r

    Where,
    n = no. of trials
    r = no. of success required
    p = probability of success in one trial
    q = probability of failure in one trial.
    n
    Cr = [n!/r!(n−r)]!
    POISSON DISTRIBUTION-
    Another probability distribution widely used
    in both the physical and social sciences to
    find out the theoretical frequencies of
    different rare events of the cases where p is
    very small.
    Examples-no. of printing mistake per page of
    a large book.
    This theory was developed by a French
    Mathematician , Simeon Denis Poisson in
    1837.
    APPLICATIONS OF POISSON
    DISTRIBUTION-
    1) Number of trials are too large (tends to
    infinity)
    2) Probability of success is very
    small(tends to Zero) (i.e. p).
    UTILITY OR IMPORTANCE OF
    POISSON DISTRIBUTION-
    1) The number of telephone calls arriving at a telephone switch board in unit time
    (say, per minute).
    2) The number of customers arriving at the super market ; say per hour.
    3) The number of defects per unit of manufactured product [This is done for the
    construction of control chart for number of defects (c) in Industrial Quality
    Control].
    4) To count the number of radio-active disintegrations of a radio-active element per
    unit of time (Physics).
    5) To count the number of bacteria per unit (Biology).
    6) The number of defective material say, pins, blades etc. in a packing
    manufactured by a good concern.
    7) The number of suicides reported in a particular day or the number of causalities
    (persons dying) due to a rare disease such as heart attack or cancer or snake bite
    in a year.
    8) The number of accidents taking place per day on a busy road.
    9) The number of typographical errors per page in a typed material or the number
    of printing mistakes per page in a book.
    FUNCTION OF POISSON
    DISTRIBUTION-

    P= f(x)= (e-λ λx )/x!
    Where,
     λ= np (Average)
    x(r)= no. of success required
    If not given,
    e= 2.71828
    PROPERTIES OF POISSON
    DISTRIBUTION-
    1) Mean= m= np
    2) Variance= m
    3) Standard deviation= √m
    4) Mode= m
    i) Integer- Bi-Modal- m &m-1
    ii) Non- Integer- Greatest Integer in m
    NORMAL DISTRIBUTION
    Normal distribution, also
    called Gaussian distribution, the most
    common distribution function for
    independent, randomly generated
    variables. Its familiar bell-shaped curve is 
    ubiquitous in statistical reports, from
    survey analysis and quality control to
    resource allocation.
    PROPERTIES OF NORMAL
    DISTRIBUTION
    1) Mean= Median = Mode
    2) Mean Deviation from Mean =
    Mean Deviation from Median=
    Mean Deviation from Mode = 0.8*standard deviation
    3) Quartile- 1st Quartile=
    µ- 0.675*standard deviation
    2nd Quartile =
    µ+0.675*standard deviation
    4) Quartile Deviation = 0.675*Standard Deviation
    5) Point of Inflexion = (µ- standard deviation) (µ+standard deviation)
    6) Normal Distribution curve is a bell shaped curve.
    7) The curve is symmetrical about the line X = μ, (Z = 0), i.e.., it has the same shape on either side of
    the line X = μ (or Z = 0). This is because the equation of the curve φ (z) remains unchanged if we
    change z to – z.
    8) Theoretically, the range of the distribution is from – ∞ to ∞. But practically, Range = 6σ
    9) Distribution is unimodal, the only mode occurring at X = μ.
    10) Quartile deviation (Q.D.) is given by Q.D. = Q3 – Q1 2 = 0·6745σ .
    11) Q.D. : M.D. : S.D. : : 10 : 12 : 15
    12) No portion of the curve lies below the x-axis, since p(x) being the probability can never be negative
    IMPORTANCE OF NORMAL
    DISTRIBUTION-
    1) Most of the discrete probability distributions (e.g., Binomial distribution, Poisson distribution) tend to
    normal distribution as n, the number of trials increases. For large values of n, computation of probability
    for discrete distributions becomes quite tedious and time consuming. In such cases, normal approximation
    can be used with great ease and convenience.
    2) Almost all the exact sampling distributions, e.g., Student’s t-distribution, Snedecor’s F-distribution,
    Fisher’s Z-distribution and Chi square distribution conform to normal distribution for large degrees of
    freedom (i.e., as n → ∞).
    3) The whole theory of exact sample (small sample) tests, viz., t, F, χ2 tests, etc., is based on the fundamental
    assumption that the parent population from which the samples have been drawn follows Normal
    distribution.
    4) Normal distribution is used in Statistical Quality Control in Industry for the setting of control limits for the
    construction of control charts.
    5) W.J. Youden of the National Bureau of Standards describes the importance of the Normal distribution
    artistically in the following words : THE NORMAL LAW OF ERROR STANDS OUT IN THE
    EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALISATIONS OF NATURAL
    PHILOSOPHY. IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES, IN THE PHYSICAL
    AND SOCIAL SCIENCES AND IN MEDICINE, AGRICULTURE AND ENGINEERING. IT IS AN
    INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA
    OBTAINED BY OBSERVATION AND EXPERIMENT.
    6) The above presentation, strikingly enough gives the shape of the normal probability curve.
    7) Lipman reveals the popularity and importance of normal distribution in the following quotation : “Every
    body believes in the law of errors (the normal curve), the experimenters because they think it is a
    mathematical theorem, the mathematicians because they think it is an experimental fact.
    PROBABILITY DENSITY FUNCTION
    OF NORMAL DISTRIBUTION-
    NORMAL DISTRIBUTION CURVE-
    z= x-µ/σ
    Where,
    x= required value(will be decided by the
    constraints)
    µ=mean
    σ= standard deviation
    And z represents area.

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