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Lecture 8 Dustin Lueker Experiment Random (or Chance) Experiment Outcome Sample Space Event Simple Event ◦ Any activity from which an outcome, measurement, or other such result is obtained ◦ An experiment with the property that the outcome cannot be predicted with certainty ◦ Any possible result of an experiment ◦ Collection of all possible outcomes of an experiment ◦ A specific collection of outcomes ◦ An event consisting of exactly one outcome STA 291 Fall 2009 Lecture 8 2 Let A and B denote two events Complement of A ◦ All the outcomes in the sample space S that do not belong to the even A ◦ P(Ac)=1-P(A) Union of A and B ◦ A∪B ◦ All the outcomes in S that belong to at least one of A or B Intersection of A and B ◦ A∩B ◦ All the outcomes in S that belong to both A and B STA 291 Fall 2009 Lecture 8 3 Let A and B be two events in a sample space S ◦ P(A∪B)=P(A)+P(B)-P(A∩B) A and B are Disjoint (mutually exclusive) events if there are no outcomes common to both A and B ◦ A∩B=Ø Ø = empty set or null set ◦ P(A∪B)=P(A)+P(B) STA 291 Fall 2009 Lecture 8 4 Can be difficult Different approaches to assigning probabilities to events ◦ Subjective ◦ Objective Equally likely outcomes (classical approach) Relative frequency STA 291 Fall 2009 Lecture 8 5 Relies on a person to make a judgment as to how likely an event will occur Events of interest are usually events that cannot be replicated easily or cannot be modeled with the equally likely outcomes approach ◦ As such, these values will most likely vary from person to person The only rule for a subjective probability is that the probability of the event must be a value in the interval [0,1] STA 291 Fall 2009 Lecture 8 6 The equally likely approach usually relies on symmetry to assign probabilities to events ◦ As such, previous research or experiments are not needed to determine the probabilities Suppose that an experiment has only n outcomes The equally likely approach to probability assigns a probability of 1/n to each of the outcomes Further, if an event A is made up of m outcomes then P(A) = m/n STA 291 Fall 2009 Lecture 8 7 Borrows from calculus’ concept of the limit a P( A) lim n n ◦ We cannot repeat an experiment infinitely many times so instead we use a ‘large’ n Process Repeat an experiment n times Record the number of times an event A occurs, denote this value by a Calculate the value of a/n a P( A) n STA 291 Fall 2009 Lecture 8 8 X is a random variable if the value that X will assume cannot be predicted with certainty ◦ That’s why its called random Two types of random variables ◦ Discrete Can only assume a finite or countably infinite number of different values ◦ Continuous Can assume all the values in some interval STA 291 Fall 2009 Lecture 8 9 Are the following random variables discrete or continuous? ◦ X = number of houses sold by a real estate developer per week ◦ X = weight of a child at birth ◦ X = time required to run 800 meters ◦ X = number of heads in ten tosses of a coin STA 291 Fall 2009 Lecture 8 10 A list of the possible values of a random variable X, say (xi) and the probability associated with each, P(X=xi) ◦ All probabilities must be nonnegative ◦ Probabilities sum to 1 0 P( xi ) 1 P( x ) 1 i STA 291 Fall 2009 Lecture 8 11 X 0 1 2 3 4 P(X) .1 .2 .2 .15 .1 5 6 7 .05 .05 .15 The table above gives the proportion of employees who use X number of sick days in a year ◦ An employee is to be selected at random Let X = # of days of leave P(X=2) = P(X≥4) = P(X<4) = P(1≤X≤6) = STA 291 Fall 2009 Lecture 8 12 Expected Value (or mean) of a random variable X ◦ Mean = E(X) = μ = ΣxiP(X=xi) Example X 2 4 6 8 10 12 P(X) .1 .05 .4 .25 .1 .1 ◦ E(X) = STA 291 Fall 2009 Lecture 8 13 Variance ◦ Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi) Example X 2 4 6 8 10 12 P(X) .1 .05 .4 .25 .1 .1 ◦ Var(X) = STA 291 Fall 2009 Lecture 8 14 A random variable X is called a Bernoulli r.v. if X can only take either the value 0 (failure) or 1 (success) Heads/Tails Live/Die Defective/Nondefective ◦ Probabilities are denoted by P(success) = P(1) = p P(failure) = P(0) = 1-p = q ◦ Expected value of a Bernoulli r.v. = p ◦ Variance = pq STA 291 Fall 2009 Lecture 8 15 Suppose we perform several, we’ll say n, Bernoulli experiments and they are all independent of each other (meaning the outcome of one even doesn’t effect the outcome of another) ◦ Label these n Bernoulli random variables in this manner: X1, X2,…,Xn The probability of success in a single trial is p The probability of success doesn’t change from trial to trial We will build a new random variable X using all of these Bernoulli random variables: n X X1 X 2 Xn Xi i 1 ◦ What are the possible outcomes of X? What is X counting? STA 291 Fall 2009 Lecture 8 16 The probability of observing k successes in n independent trails is n k nk P( X k ) p q , k 0,1, k , n, ◦ Assuming the probability of success is p ◦ Note: n n! k k!(n k )! Why do we need this? STA 291 Fall 2009 Lecture 8 17 For small n, the Binomial coefficient “n choose k” can be derived without much mathematics n n! k k !(n k )! Example: where n ! 1 2 3 n and 0! 1 4 4! 4! 1 2 3 4 6 2 2!(4 2)! 2! 2! 1 2 1 2 STA 291 Fall 2009 Lecture 8 18 Assume Zolton is a 68% free throw shooter ◦ What is the probability of Zolton making 5 out of 6 free throws? 6 P ( X 5) 0.685 (1 0.68) 65 5 6 0.1454 0.32 0.279 ◦ What is the probability of Zolton making 4 out of 6 free throws? 6 4 6 4 P( X 4) 0.68 (1 0.68) 4 15 0.2138 0.1024 0.3284 STA 291 Fall 2009 Lecture 8 19 n p n p (1 p) 2 n p (1 p) STA 291 Fall 2009 Lecture 8 20