Section 7.1.1Discrete and Continuous Random VariablesAP Statistics

AP Statistics, Section 7.1, Part 1 2

Random Variables A random variable is a variable whose value is

a numerical outcome of a random phenomenon. For example: Flip three coins and let X

represent the number of heads. X is a random variable.

We usually use capital letters to denotes random variables.

The sample space S lists the possible values of the random variable X.

We can use a table to show the probability distribution of a discrete random variable.

AP Statistics, Section 7.1, Part 1 3

Discrete Probability Distribution Table

Value of X: x1 x2 x3 … xn

Probability:p1 p2 p3 … pn

AP Statistics, Section 7.1, Part 1 4

Discrete Random Variables

A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities.

X: x1 x2 x3 … xk

P(X): p1 p2 p3 … pk

1. 0 ≤ pi ≤ 1

2. p1 + p2 + p3 +… + pk = 1.

AP Statistics, Section 7.1, Part 1 5

Probability Distribution Table: Number of Heads Flipping 4 Coins

TTTT

TTTH

TTHT

THTT

HTTT

TTHH

THTH

HTTH

HTHT

THHTHHTT

THHH

HTHH

HHTH

HHHT

HHHH

X 0 1 2 3 4

P(X) 1/16 4/16 6/16 4/16 1/16

AP Statistics, Section 7.1, Part 1 6

Probabilities: X: 0 1 2 3 4 P(X): 1/16 1/4 3/8 1/4 1/16

.0625 .25 .375 .25 .0625 Histogram

AP Statistics, Section 7.1, Part 1 7

Questions.

Using the previous probability distribution for the discrete random variable X that counts for the number of heads in four tosses of a coin. What are the probabilities for the following?

P(X = 2) P(X ≥ 2) P(X ≥ 1)

.375

.375 + .25 + .0625 = .6875

1-.0625 = .9375

AP Statistics, Section 7.1, Part 1 8

What is the average number of heads?

61 4 4 116 16 16 16 16

0 4 12 12 416 16 16 16 16

3216

0 1 2 3 4

2

x

AP Statistics, Section 7.1, Part 1 9

Continuous Random Varibles Suppose we were to randomly generate a

decimal number between 0 and 1. There are infinitely many possible outcomes so we clearly do not have a discrete random variable.

How could we make a probability distribution? We will use a density curve, and the probability

that an event occurs will be in terms of area.

AP Statistics, Section 7.1, Part 1 10

Definition:

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 and above the values of X that make up the event.

All continuous random distributions assign probability 0 to every individual outcome.

AP Statistics, Section 7.1, Part 1 11

Distribution of Continuous Random Variable

AP Statistics, Section 7.1, Part 1 12

AP Statistics, Section 7.1, Part 1 13

Example of a non-uniform probability distribution of a continuous random variable.

AP Statistics, Section 7.1, Part 1 14

Problem

Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is a uniform distribution.

Draw the density curve. What is the probability that the wait is

between 12 and 20 minutes?

AP Statistics, Section 7.1, Part 1 15

Density Curve.

20151050

0.05

0.04

0.03

0.02

0.01

0.00

X

Densi

ty

Distribution PlotUniform, Lower=0, Upper=20

AP Statistics, Section 7.1, Part 1 16

Probability shaded.

0.05

0.04

0.03

0.02

0.01

0.00

X

Densi

ty

12

0.4

200

Distribution PlotUniform, Lower=0, Upper=20

P(12≤ X ≤ 20) = 0.5 · 8 = .40

AP Statistics, Section 7.1, Part 1 17

Normal Curves

We’ve studied a density curve for a continuous random variable before with the normal distribution.

Recall: N(μ, σ) is the normal curve with mean μ and standard deviation σ.

If X is a random variable with distribution N(μ, σ), then is N(0, 1)X

Z

AP Statistics, Section 7.1, Part 1 18

Example Students are reluctant to report cheating by

other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?”

Suppose that if we could ask all undergraduates, 12% would answer “Yes.” The proportion p = 0.12 would be a parameter for the population of all undergraduates.

p̂

0.016). N(0.12, ofon distributi a with variablerandom a is ˆ . estimate toused

statistic a is yes""answer whosample theof ˆ proportion The

pp

p

AP Statistics, Section 7.1, Part 1 19

Example continued

Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?”

What is the probability that the survey results differs from the truth about the population by more than 2 percentage points?

Because p = 0.12, the survey misses by more than 2 percentage points if .14.0ˆor 10.0ˆ pp

AP Statistics, Section 7.1, Part 1 20

AP Statistics, Section 7.1, Part 1 21

Example continued Calculationsˆ ˆ ˆ( 0.10 or 0.14) 1 (0.10 0.14)

From Table A,

ˆ0.10 0.12 0.12 0.14 0.12ˆ(0.10 0.14)

0.016 0.016 0.016

( 1.25 1.25)

0.8944 0.1056 0.7888

So,

ˆ ˆ( 0.10 or 0.14) 1 0.7888 0.2112

P p p P p

pP p P

P Z

P p p

About 21% of sample results will be off by more than two percentage points.

AP Statistics, Section 7.1, Part 1 22

Summary

A discrete random variable X has a countable number of possible values.

The probability distribution of X lists the values and their probabilities.

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 and above the values of X that make up the event.

AP Statistics, Section 7.1, Part 1 23

Summary

When you work problems, first identify the variable of interest.

X = number of _____ for discrete random variables.

X = amount of _____ for continuous random variables.