Section 7.1.1

Discrete and Continuous Random Variables

AP Statistics

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.

2

AP Statistics, Section 7.1, Part 1

Discrete Probability Distribution Table

Value of X: x 1 x 2 x 3 … x n Probability: p 1 p 2 p 3 … p n AP Statistics, Section 7.1, Part 1

3

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: x 1 P(X):

p 1

x

p

2

2

x 3

p 3

… x k

… p k

1. 0 ≤

p i ≤ 1

2.

p 1 + p 2 + p 3 +… + p k = 1.

4

Probability Distribution Table: Number of Heads Flipping 4 Coins

X TTTT 0 TTTH TTHT THTT HTTT 1 TTHH THTH HTTH HTHT THHT HHTT 2 THHH HTHH HHTH HHHT 3 HHHH 4 P(X) 1/16 4/16 6/16 AP Statistics, Section 7.1, Part 1 4/16 1/16

5

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

6

AP Statistics, Section 7.1, Part 1

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

7

AP Statistics, Section 7.1, Part 1

x

What is the average number of heads?

0 16 16 1 16 4 16 1 4 16 12 16 2 2 12 16 16 4 16 3 4 16 4 1 16 AP Statistics, Section 7.1, Part 1

8

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.

9

AP Statistics, Section 7.1, Part 1

Definition:

 A

continuous random variable X

values in an interval of numbers. takes all  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.

10

AP Statistics, Section 7.1, Part 1

Distribution of Continuous Random Variable

AP Statistics, Section 7.1, Part 1

11

AP Statistics, Section 7.1, Part 1

12

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

AP Statistics, Section 7.1, Part 1

13

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?

14

AP Statistics, Section 7.1, Part 1

Density Curve.

Distribution Plot

Uniform, Lower=0, Upper=20 0.05

0.04

0.03

0.02

0.01

0.00

0 5 10 15 20

15

Probability shaded.

Distribution Plot

Uniform, Lower=0, Upper=20 0.05

0.04

0.03

0.02

0.01

0.00

0

X

12 P(12 ≤ X ≤ 20) = 0.5 · 8 = .40

AP Statistics, Section 7.1, Part 1 0.4

20

16

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.

Z



X

   μ, σ),

17

AP Statistics, Section 7.1, Part 1

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?”

p

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.

The proportion

p

ˆ of the sample who answer " yes" is a statistic used to estimate

p

.

ˆ

p

is a random variable with a distributi on of N(0.12, 0.016).

18

AP Statistics, Section 7.1, Part 1

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  0 .

10 or  0 .

14 .

19

AP Statistics, Section 7.1, Part 1

AP Statistics, Section 7.1, Part 1

20

Example continued Calculations

(  0.10 or  0.14) From Table A,

P

(0.10

0.14)

P

(0.10  

P

  0.14) 

P

0.016

1.25)  0.7888

( So,  0.10 or  0.14)   0.12

0.016

 0.2112

 0.016

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

AP Statistics, Section 7.1, Part 1

21

Summary

    A

discrete random variable X

number of possible values. has a countable The

probability distribution

values and their probabilities.

of X lists the A

continuous random variable X

values in an interval of numbers. takes all 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.

22

AP Statistics, Section 7.1, Part 1

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.

23

AP Statistics, Section 7.1, Part 1