Finals - Stat 115

• Population - the collection of all elements under consideration in a statistical inquiry.
• Sample - a subset of individuals selected from a larger group.
• Parameter describes a specific characteristic of the population.
• Statistic describes a specific characteristic of the sample.
• Both the parameter and statistic are summary measures that are computed using data.
• Statistical Inquiry - the process that leads to the answer to the research problem.
• Descriptive Statistics - methods concerned with collecting, describing, and analyzing a set of data without drawing conclusions or inferences about a larger group.
• Inferential Statistics - methods concerned with the analysis of sample data leading to predictions or inferences about the population.
• Inferential Statistics - to estimate with certain level of confidence.
• Inferential Statistics has conditions of uncertainty due to the use of partial information, therefore, the conclusions will be subject to some error.
• Probability Theory

  • first developed to give answers to professional gambler’s questions
  • basic examples include die-throwing experiments and the selection in a deck of cards.
• Random Experiment

  • process that can be repeated under similar conditions but whose outcome cannot be predicted with certainty beforehand
  • outcomes may vary each time the process is repeated, even under the exact same conditions
• Sample Space

  • denoted by Ω (Greek letter omega)
  • collection of all possible outcomes of a random experiment
  • referred to as the universal set in set theory
• An element of the sample space is called a sample point.
• Two Methods of defining sample space

  • Roster Method
  • Rule Method
• Ordered k-tuple

  • k coordinates, where k is an integer greater than 2
  • an extension of the concept of an ordered pair
• Simple Random Sampling

  • Without Replacement (SRSWOR)
  • all possible subsets consisting of n distinct elements selected from the N elements of the population have the same chances of selection.
  • n(Ω) = NCn for cardinality
  • n(A) = n/N for inclusion probability
  • With Replacement (SRSWR)
  • all possible ordered n-tuples (coordinates need not be distinct) that can be formed from the N elements of the population have the same chances of selection
  • n(Ω) = N^n for cardinality
  • n(A) = 1-(N-1/N)^n for inclusion probability
• • combination = order is not important
  • in how many ways can you choose
  • permutation = order is important
  • in how many ways can you arrange
• Event

  • a subset of the sample space whose probability is defined
  • We say that it occurred if the outcome of the random experiment is one of the sample points belonging in the event; otherwise, it did not occur.
  • denoted by capital Latin Letters
• • Two special events
  • Ω, sure event
  • ∅, impossible event
• Mutually Exclusive Events

  • If and only if their intersection is = ∅ → A ∩ B = ∅
  • That is, A and B have no elements in common.
• Set operations

  • and/ all/ both/ but (intersection) - elements that are in both sets
  • n(A∩B) = n(A)+n(B)−n(A∪B)
  • at least/ either/ or (union) - elements contained in either set
  • n(A∪B) = n(A) + n(B) − n(A∩B)
  • complement A’ = the set of elements not in A
  • n(A’) = n(Ω) – n(A)
  • n(A∩B’) = n(A) - n(A∩B)
  • De-Morgan's Law
  • (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B’
  • AUBUC
  • n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
• Independent Events

  • P(A ∩ B) = P(A) x P(B)
  • P(A) > 0 and P(B) > 0
  • event cannot be both mutually exclusive and independent unless one event is zero
• Probability of an Event

  • The probability of an event A, denoted by P(A), is a function that assigns a measure of chance that event A will occur, and must satisfy the following properties:

  1. Nonnegativity: 0 ≤ P(A) ≤ 1
  2. Norming Axiom: P(Ω) = 1; and P(∅) = 0
  3. Finite Additivity
• Interpretation of the Probability

  • Close to 1 → very high chance of happening
  • Close to 0 → very low chance of happening
  • 0.5 → equal chances to happen or not happen
• Three Approaches to Assigning Probabilities

  • A Priori or Classical Approach
  • A Posteriori or Relative Frequency Approach
  • Subjective Approach
• A Priori or Classical Approach

  • Definition
  • (from what is earlier of the experiment)
  • If a random experiment can result in any one of N different equally likely outcomes and if exactly n of these outcomes belong in event A, then P(A) = n/N
  • allows us to view proportions in terms of probabilities and vice versa
  • P(A) = proportion of elements in the population that possess the characteristic of interest
  • probability can only take values from 0 to 1
• A Posteriori or Relative Frequency Approach

  • (from what is later of the experiment)
  • If a random experiment is repeated many times under uniform conditions, we use the empirical probability of event A to assign its probability.
  • Empirical P(A) = number of times event A occurred/ number of times experiment was repeated
  • P(A) is the limiting value of its empirical probability if we repeat the process endlessly.
  • This empirical probability will be a good approximate of the actual probability if we perform the process a large number of times under uniform conditions.
• Subjective Approach

  • The probability of occurrence of an event A is determined by the use of intuition, personal beliefs, and other indirect information.
  • denoted by the value p, where 0 ≤ p ≤ 1
  • we use this approach when the former two is not feasible to use or when the experiments done are unique
• • Random Variable - a function whose value is a real number that is determined by each sample point in the sample space
  • An uppercase letter, say X, will be used to denote a random variable and its corresponding lowercase letter, x in this case, will be used to denote one of its values.
  • Variable - the characteristic of interest whose value varies
  • the realized or actual value of the variable depends on the outcome of a random experiment.
  • it is impossible to predict with certainty what the realized value of the random variable X will be.
• The cumulative distribution function (CDF) of a random variable X, denoted by F(·) is a function defined for any real number x as F(x) = P(X ≤ x); where X is a random variable and x is a specified real number.
• The CDF of the random variable X is also referred to as its distribution.

  • non-decreasing and can take values from 0 to 1 (probability or area)
  • provides us with complete information about the behavior of the random variable.
  • We can use it to compute for the probability of any event expressed in terms of the random variable X.
• Two Major Types of Random Variables

  1. Discrete random variable
  2. (Absolutely) continuous random variable
• Discrete random variable

  • If a sample space contains a finite number of sample points or has as many sample points as there are counting or natural numbers, then it is called a discrete sample space.
  • A random variable defined over a discrete sample space is called a discrete random variable.
  • The probability mass function (PMF) of a discrete random variable (discrete probability distribution) , denoted by f(·), is a function defined for any real number x as f(x) = P(X=x)
  • The values of the discrete random variable X for which f(x)>0 are called its mass points.
• Constructing the PMF of X

  • Step 1. Identify the mass points of X.
  • The mass points of X are actually the possible values that X could take on because these are the points where P(X=x) will be nonzero. In other words, the set of mass points of X is the range of the function, X.
  • Step 2. Determine the event associated with the expression, X=x.
  • Step 3. Compute for the probability of this event.
• Using the PMF of X to Compute for Probabilities of Events Expressed in Terms of X

  • Step 1. Identify the mass points, x, that are included in the interval of interest.
  • Step 2. Use the PMF to determine the value of P(X=x) for each one of the mass points identified in Step 1.
  • Step 3. Get the sum of all the values derived in Step 2.
• Deriving the CDF of the discrete random variable X from its PMF

  • the graph looks like a staircase
  • step function
  • in half-open intervals, the graph is flat and then the value of the function suddenly jumps up
  • jumps occur at the mass points, and the value increased from the previous point is the probability
  • the value of the CDF only increases at mass points
  • the size of the increase is always equal to f(x)
  • f(x) is constant elsewhere
  • for values less than 1 (the smallest mass point), the value of the CDF is 0, for values greater than or equal to the largest mass point, the value of the CDF is 1
• (Absolutely) continuous random variable

  • For a continuous random variable, X, the P(X=x) will always be 0 for any real number x.
  • this property will be satisfied by variables that are measured using some standard measurement of scale of real numbers or nonnegative real numbers
  • Any particular measure taken on such scales could be recorded to as many decimal places as one might care to take it.
• The probability density function (PDF) of a continuous random variable X, denoted by f(·), is a function that is defined for any real number x and satisfy the following properties:

  • f(x)≥ 0
  • The area below the whole curve and above the x-axis is always equal to 1
  • P( a ≤ X ≤ b ) - The shaded area which is bounded by the curve f(x), the x-axis, and the lines x=a and x=b
• The graph of the PDF is always above the x-axis because the function cannot take on negative values.

  • P(X=a) is just the same as P( a ≤ X ≤ a ) . In this case, we will let b=a . Then, the area representing P(X=a) will be 0 because we will only be left with a single line.
  • X<a will always be equal to P(X≤ a)

  1. P(X≤ a) = P(X<a) + P(X=a) = P(X<a)
  2. P(a<X<b) = P(a≤X<b) = P(a<X≤b) = P(a≤X≤b)