Discrete and Continuous Random Variables

Section 6.1

Reference Text:

The Practice of Statistics, Fourth Edition.

Starnes, Yates, Moore

Objectives1. Discrete Random Variables

1. What is a discrete random variable?

2. Mean (Expected Value) of a DRV1. Examples: Apgar Scores of Babies, Roulette

3. Standard Deviation (and variance) of a DRV1. Calculator saves the day!

2. Continuous Random Variables1. 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 TTT

X = 1 HTT, THT, TTH

X = 2 HHT, HTH, THH

X = 3 HHH

Intro Example• We can summarize the probability

distribution of X as follows:

We just talked about a discrete random variable!

- A discrete random variable X takes a fixed set of possible values with gaps between

Value (X) : 0 1 2 3

Probability: 1/8 3/8 3/8 1/8

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.

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 1 2 3 4 5 6 7 8 9 10

Probability: .001 .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

xz

Objectives1. Discrete Random Variables

1. What is a discrete random variable?

2. Mean (Expected Value) of a DRV1. Examples: Apgar Scores of Babies, Roulette

3. Standard Deviation (and variance) of a DRV1. Calculator saves the day!

2. Continuous Random Variables1. 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