STA 291 Fall 2009

1. STA 291Fall 2009 Lecture 8 Dustin Lueker

2. Probability Terminology • Experiment • Any activity from which an outcome, measurement, or other such result is obtained • Random (or Chance) Experiment • An experiment with the property that the outcome cannot be predicted with certainty • Outcome • Any possible result of an experiment • Sample Space • Collection of all possible outcomes of an experiment • Event • A specific collection of outcomes • Simple Event • An event consisting of exactly one outcome STA 291 Fall 2009 Lecture 8

3. Basic Concepts • 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

4. Probability • 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

5. Assigning Probabilities to Events • 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

6. Subjective Probability Approach • 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

7. Equally Likely Approach • 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

8. Relative Frequency Approach • Borrows from calculus’ concept of the limit • 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 STA 291 Fall 2009 Lecture 8

9. Random Variables • 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

10. Examples • 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

11. Discrete Probability Distribution • 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 STA 291 Fall 2009 Lecture 8

12. Example • 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

13. Expected Value of a Discrete Random Variable • Expected Value (or mean) of a random variable X • Mean = E(X) = μ = ΣxiP(X=xi) • Example • E(X) = STA 291 Fall 2009 Lecture 8

14. Variance of a Discrete Random Variable • Variance • Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi) • Example • Var(X) = STA 291 Fall 2009 Lecture 8

15. Bernoulli Random Variables • 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

16. Binomial Distribution • 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 nBernoulli 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: • What are the possible outcomes of X? What is X counting? STA 291 Fall 2009 Lecture 8

17. Binomial Distribution • The probability of observing k successes in n independent trails is • Assuming the probability of success is p • Note: • Why do we need this? STA 291 Fall 2009 Lecture 8

18. Binomial Coefficient • For small n, the Binomial coefficient “n choose k” can be derived without much mathematics STA 291 Fall 2009 Lecture 8

19. Example • Assume Zolton is a 68% free throw shooter • What is the probability of Zolton making 5 out of 6 free throws? • What is the probability of Zolton making 4 out of 6 free throws? STA 291 Fall 2009 Lecture 8

20. Binomial Distribution Properties STA 291 Fall 2009 Lecture 8