Discrete and Continuous Random Variables Section 6.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Objectives 1. Discrete Random Variables 1. What is a discrete random variable? 2. Mean (Expected Value) of a DRV 1. Examples: Apgar Scores of Babies, Roulette 3. Standard Deviation (and variance) of a DRV 1. Calculator saves the day! 2. Continuous Random Variables 1. What is a continuous random variable? 1. Area under the curve! 2. Finding the probability of the interval of outcomes, Z-scores return! Intro Example: • Suppose we toss a fair coin 3 times. The sample space for this chance process is: • HHH HHT HTH THH HTT THT TTH TTT • Since there are 8 equally likely outcomes the probability is 1/8 for each possible outcome. • Define the variable X = the number of heads obtained. • What are my outcomes of possible heads? X=0 X=1 X=2 X=3 TTT HTT, THT, TTH HHT, HTH, THH HHH Intro Example • We can summarize the probability distribution of X as follows: Value (X) : 0 1 2 3 Probability: 1/8 3/8 3/8 1/8 We just talked about a discrete random variable! - A discrete random variable X takes a fixed set of possible values with gaps between What is A Discrete Random Variable • We have learned several rules of probability but one way of assigning probabilities to events: assign probabilities to every individual outcome, then add these probabilities to find the probability of any event. • This idea works well if we can find a way to list all possible outcomes. We will call random variables having probability assigned in this way discrete random variables. Value: ….. Probability: ….. Requirements of DRV Apgar Scores: Babies’ Health at Birth • In 1952, Dr. Virginia Apgar suggested five criteria for measuring a baby’s health at birth: skin color, heart rate, muscle tone, breathing, and response when stimulated. She developed a 0-1-2 scale to rate a newborn on each of the five criteria. A baby’s Apgar score is the sum of the ratings on each of the five scales, which gives a whole-number value from 0 to 10. Apgar scores are still used today to evaluate the health of newborns. Apgar Scores: Babies’ Health at Birth • What Apgar scores are typical? To find out, researchers recorded the Apgar scores of over 2 million newborn babies in a single year. Imagine selecting one of these newborns at random. (that’s our chance process). Define the random variable X = Apgar score of a randomly selected baby one minute after birth. The table below gives the probability distribution for X. Value 0 Probability: .001 1 2 3 4 5 6 7 8 9 10 .006 .007 .008 .012 .020 .038 .099 .319 .437 .053 Apgar Scores: Babies’ Health at Birth A) Show that the probability distribution for X is legitimate B) Doctors decided that Apgar scores of 7 or higher indicate of healthy baby. What's the probability that a randomly selected baby is healthy. Mean (Expected Value) Of A Discrete Random Variable Winning (and losing) at Roulette • On an American roulette wheel, there are 38 slots numbered 1 through 36, plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a simple $1 bet on red. If the ball lands on a red slot, the player gets the original dollar back, plus an additional dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the dollar bet to the casino. • Lets define the random variable X = net gain from a single $1 bet on red. The possible values of X are -$1 and $1 (the player either gains a dollar or loses a dollar.) What are the corresponding probabilities? The chance that the ball lands on red slot is 18/38. The chance that the ball lands in a different-colored slot is 20/38. Here is the probability distribution of X: Value -$1 $1 Probability: 20/38 18/38 Mean (Expected Value) Of A Discrete Random Variable Find the Mean for Apgar Scores! Standard Deviation (and Variance) for a DRV Standard Deviation (and Variance) for a DRV Find the Standard Deviation for Apgar Scores! That was hard! • Good thing we have a calculator to help reduce time consumption! • TI-83 – Start by entering the values of X in L1, and probability in L2 – 1-var Stats L1, L2 • TI-89 – In the Statistics/List Editor, press F4 (calc) and choose 1:1-var stats…use the inputs list: list1 and freq: list 2 Continuous Random Variables • What if there were infinite probabilities? We cant add them all up! • So we look at the area under the curve! • Why? Well the area under the curve is 1, and probability adds up to 1, so the area under the curve can also represent the probability. • Difference: We cant look at individual probabilities…we have to look at an interval! • In fact, all continuous probability models assign probability 0 to every individual outcome. Continuous Random Variables • A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve. • Lets look at an example of finding the probability! Young Women’s Heights z x Objectives 1. Discrete Random Variables 1. What is a discrete random variable? 2. Mean (Expected Value) of a DRV 1. Examples: Apgar Scores of Babies, Roulette 3. Standard Deviation (and variance) of a DRV 1. Calculator saves the day! 2. Continuous Random Variables 1. What is a continuous random variable? 1. Area under the curve! 2. Finding the probability of the interval of outcomes, Z-scores return! Homework Worksheet