Probability Review 1 Probability Theory Mathematical description of

Probability Review 1

Probability Theory • Mathematical description of relationships or occurrences that cannot be predicted precisely • An experiment is an activity whose outcome is subject to random (i. e. chance or unknown) variation. Examples: – Flip a coin – Toss a die – … 2

Sample Space • The set of all possible outcomes of an experiment is known as its sample space, denoted as S Examples: – Coin flipping S={ } – Tossing a die S={ } 3

Events • An event is a collection of outcomes from a sample space, denoted as E Examples: – Coin flipping Event = Get a tail – Tossing a die Event = Get an even number : outcome = {2, 4, 6} • An event E is said to occur if one of the outcomes with which it is associated is realized during a replication of the experiment 4

Events • Two events, E and F, are said to be mutually exclusive if E∩F= which means… • The complement of event E, denoted Ec, is that unique set such that E U Ec = S and E ∩ Ec = 5

Random Variable • A random variable is a function that maps the sample space to the real line. Examples: – Coin flipping X= 1 if heads 0 if tails – Tossing a die W = (the number that shows on die) • A random variable is discrete if the possible values it can assume can be counted • A random variable is continuous if it can assume any value in a continuous subset of the real line 6

Probability • The probability associated with a particular event E, denoted P(E), can be thought of as representing the relative likelihood of that event occurring • We will be generally thinking in terms of the probability of a random variable taking a specific value Examples: – Coin flipping P(X=1) = – Tossing a die P(W=6) = P(W=7) = 7

Axioms of Probability • 0 ≤ P(E) ≤ 1 for any E • P(S) = 1 • If {Ei, i=1, …, k} are mutually exclusive events, then 8

Probability Distributions • Describes probabilities of values a random variable could take • Discrete Examples: – Coin flipping P(X=x) = ½ if x={0, 1} 0 otherwise – Tossing a die P(W=w) = 1/6 if w={1, 2, 3, 4, 5, 6} 0 otherwise • Continuous Examples: – Altitude of an airplane Area under curve = 9

Common Probability Distributions • Discrete – – – Discrete uniform Poisson Geometric Binomial … • Continuous – – – – Uniform Exponential Normal Gamma Beta Triangular … 10

PDF(PMF) vs. CDF • Probability density function (p. d. f. ) denote f • Probability mass function (p. m. f. ) denote f f(x)=P(X=x) • Cumulative distribution function (c. d. f. ) denote F F(x)=P(X≤x) 11

Mean and Variance • Mean (Expected Value) E[X] = = or • Variance (Expected square distance from mean) Var(X) = 2 = E[(X-E[X])2] = E[X 2] – E[X]2 • Standard deviation (Spread) 12

Examples of Mean and Variance • Coin flipping – Expected value = – Variance = – Standard deviation = • Tossing a die – Expected value = – Variance = – Standard deviation = • Continuous uniform between 0 and 2 – Expected value = – Variance = – Standard deviation = 13

Conditional Probabilities • Consider two experiments with S 1={E 1, …, Em} and S 2={F 1, …, Fn} • P(E|F) = P{experiment 1 gets outcome E given that experiment 2 gets outcome F} • Example: P(Ice cream sales > 10 cones | temperature = 85 F) 14

Example 1 of Conditional Probabilities • The king comes from a family of 2 children. What is the probability that the other child is his sister? 15

Example 2 of Conditional Probabilities • 52% of the students at a certain college are females. 5% of the students in this college are majoring in computer science. 2% of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that 1) this student is female, given that the student is majoring in computer science; 2) this student is majoring in computer science, given that the student is female. 16

Bayes’ Theorem 17

Example 1 of Bayes’ Theorem • Suppose that an insurance company classifies people into one of three classes – good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1 -year span are, respectively, 0. 05, 0. 15, and 0. 30. If 20% of the population are “good risks”, 50% are “average risks”, and 30% are “bad risks”, what proportion of people have accidents in a fixed year? If policy holder A had no accidents in 1987, what is the probability that he or she is a good risk? 18

Example 2 of Bayes’ Theorem • Suppose that there was a cancer diagnostic test that was 95% accurate both on those that do and those that do not have the disease. If 0. 4% of the population have a cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so. 19