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ES03 - Coggle Diagram

- - - - Probability Distributions of a RV
        
        define a function that provides the probability that X is in the interval [a,b]
        
        Probability Density Function (pdf)
        
        1 more item...
        
        for two dependent RVs
        
        Joint Probability Density Function
        
        define a function that provides the probability that X is less than "a"
        
        Cumulative Distribution Function (cdf)
        
        summarized by
        
        Statistics
        
        Expected Value / Mean of RV
        
        Variance of RV
        
        Standard Deviation of a RV
        
        define a function that specifies the probability that X takes each discrete values
        
        Probability Mass Function (pmf)
        
        for two dependent RVs
        
        Joint Probability Mass Function
        
        simplified to a product of probabilities if RVs are independent
    - - The probability that a measurement belongs to the set of real number is 1, that of belonging to any set is between 0 and 1, and that of belonging to the union of several sets is the sum of the probabilities that it belongs to each set, provided they are mutually exclusive.
    - - The probability that a result falls into the set of all possible outcomes is 1, that of falling into any set of outcomes is between 0 and 1, and that of falling into the union of several events is the sum of the probabilities that it falls in each event, provided they are mutually exclusive.
  - - - Discrete RV
        
        Binomial RV
        
        Binomial experiment
        
        each trial results in only two outcomes (success or failure)
        
        probability of success on each trial remains constant
        
        n repeated trials that are independent
        
        is the number of successes in n repeated trials of a binomial experiment
        
        Binomial and Poisson distributions can be approximated to Normal distributions
        
        Poisson RV
        
        Poisson process
        
        Partition the interval T of real numbers into subintervals of
        length dt and assume that as dt tends to zero:
        
        every subinterval has the same probability of one event
        
        the probability of more than one event in a subinterval tends to zero (Bernoulli trial)
        
        the event in each subinterval is independent of other subintervals
        
        is the number of events occurring randomly in an interval
    - - 3.5 Continuous RV
        
        Normal R.V.
        
        e.g. the average results of random experiments that are replicated a number of times
        
        Lognormal RV
        
        e.g. the lifetime of a product that degrades over time
        
        Gamma RV
        
        it's used in interval estimation and tests of hypothesys
        
        Weibull RV
        
        e.g. the time until failure of physical systems
        
        Beta RV
        
        e.g. the proportion of radiation absorbed by a material, or the proportion of the total time needed to complete a task. In all these cases, X is in [0,1]
        
        given the data, which probability distribution?
        
        Probability Plots
        
        probability paper available for each distribution
        
        Exponential RV
        
        is the distance between successive events of a Poisson Process
  - - - simplified if the RVs are independent
    - - functions of a the RVs in a random sample
      - model data as observation from independent RVs with the same distribution i.e. a random sample
      - Mean
        
        Central Limit Theorem