Data and Modeling

• It is important to distinguish random from systematic variation because it can help understand the processes
• The problem of randomness cannot be eliminated, but it can be understood through probability and stochastic thinking
• A causal relationship can explain the world completely
• We can only say that something is likely to occur
• Probability


  The fraction of the number of desired outcomes over all outcomes in an experiment or observation
• Total probability


  The fundamental rule relating marginal probabilities to conditional probabilities
• Joint probability

  The likelihood of two events occurring together and at the same point in time
• Random variable


  A variable taking on numerical values determined by the outcome of a random phenomenon
• Statistical analysis attempts to separate the signal in the data from the noise
• Random variation


  Variability of a process caused by many irregular fluctuations or chance factors that cannot be anticipated, detected, identified, or eliminated
• Determinism


  All events are completely determined by previously existing causes
• Discrete uniform distribution


  A symmetric probability distribution where a finite number of values of X are equally likely to be observed
• Likelihood


  The probability distributions can calculate the likelihood of a value
• Probability distribution


  The function of variable X, evaluated at x, is the probability that X will take a value equal to x
• Geometric distribution


  The probability distribution of the number of trials needed to get one success with a probability of p
• Bayes theorem


  Calculates conditional probabilities and combines subjective or prior knowledge with objective current info to derive meaningful outcomes
• Certainty is usually unjustified, but uncertainty makes us uncomfortable
• Probability


  Can take values between 0 and 1, where 0 is impossible and 1 is certain
• Stochastic thinking


  Involves probability
• Conditional probability


  The measure of the probability of an event occurring, given that another event has already occurred
• Random variable


  Value is unknown or a function assigns the value
• Discrete variable

  A variable with a finite range, usually integer counts
• There are two alternatives in the Geometric distribution: one deals with the number of trials and the other deals with the number of failures
• Bernoulli distribution


  The probability distribution of a random variable which takes the value "1" with probability p and the value "0" with probability q=1-p
• Cumulative distribution


  The function of variable X, evaluated at x, is the probability that X will take a value less than or equal to x
• Natural or unnatural phenomena usually have random variation
• Probability


  The proportion of times an event occurs in a long run sequence or number of trials
• Independence of events

  Events are independent if the occurrence of one does not affect the probability of occurrence of the other
• Conditional independence


  Two random events A and B are conditionally independent given a third event C
• Events can occur multiple times (N times)
• The world is possibly inherently unpredictable, and we do not have all the knowledge to make accurate predictions
• Cause will always have an effect
• Discrete distributions

  • Discrete uniform distribution
  • Bernoulli distribution
  • Binomial distribution
  • Geometric distribution
• Random variable

  Can be either discrete or continuous
• Random variation


  The sum of many small variations inherent in a process, which cannot be tracked back to a root cause
• Binomial distribution


  The probability distribution of the number of successes in a sequence of n independent experiments with probability p
• Continuous variable


  A variable that can take infinitely many values within some interval of numbers
• Geometric distribution


  • Deals with number of trials
  • Deals with number of failures
  • Useful for assessing reliability and survival analysis
• Continuous uniform distribution


  Symmetric probability distribution describing an experiment where outcomes lie between certain boundaries
• Log-normal distribution

  Continuous probability distribution of a random variable whose logarithm is normally distributed, useful for variables that cannot be negative