Statistics Lecture 7

Last day: Completed Chapter 2 Today: Discrete probability distributions Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110

Example (Chapter ) A system of components is connected as in the following diagram. Components 1 and 2 are connected in parallel so the subsystem works if and only if either 1 or 2 works Components 3 and 4 are connected in series, so the subsystem works iff and only iff both 3 and 4 work Assume that the system works iff either the first or second subsystem works Assume that the components work independently of each other and P(Component works)=0.9 Find P(system works)

Example (Chapter )

Example – Let’s Make a Deal: A contestant is given a choice of three doors of which one contained a prize such as a Car The other two doors contained gag gifts like a chicken or a donkey After the contestant choses an initial door, the host of the show reveals an empty door among the two unchosen doors, and asks the contestant if they would like to switch to the other unchosen door What should the contestant do?

Example Roll two dice Events: A 1 ={first die response is odd} A 2 ={second die response is odd} A 3 ={Sum of dice is odd} Are the events mutually independent?

Another Example N people go to a restaurant and check their coats The coats are given back randomly What is the probability that no one receives their own coat

Chapter 3 – Discrete Random Variables Recall – an experiment is a process where the outcome is uncertain The experiment can take on a variety of outcomes Each outcome can be associated with a number by specifying a rule of association….a random variable is such a rule Random Variable: For a given sample space, S, a random variable is any rule that associates a number with each outcome in S Can be viewed as a map from the sample space to the real line

Will consider two types: Discrete random variables Continuous random variables

Discrete versus Continuous Discrete random variables have either a finite number of values or infinitely many values that can be ordered in a sequence Continuous random variables take on all values in some interval(s)

Examples Discrete or continuous Number of people arriving in a supermarket Hair color of randomly selected people Weight lost from a diet program Random number between 0 and 4

Discrete Random Variables Describe chances of observing values for a discrete random variable by probability distribution or probability mass function Probability distribution of a discrete random variable, X, is the list of distinct numerical outcomes and associated probabilities P(X=x i )=p(x i )

Example Flip a coin Get responses heads or tails S={H,T} X(H)=1X(T)=0 Random variable X takes on value 1 for heads and 0 for tails A rv that takes on two values, 0 and 1, is called a Bernoulli rv

Properties for each value x i of X

Example Consider a baseball player with a 300 batting average (i.e., gets hit 30% of the time) Let X be the number of at bats until the batter gets a hit Describe the probability distribution for X

Can display distribution using a probability histogram X-axis represents outcomes Y-axis is the probability of each outcome Use rectangles, centered at each value of X, to display probabilities

Example Probability distribution for number people in a randomly selected household Draw the probability histogram