5.1 Random Variables and Probability Distributions

5.1 Random Variables and Probability Distributions

Statistical Experiment

A statistical experiment is any process by which an observation or a

measurement is made.

Example A

Statistical Experiment

a. Measure the daily rainfall in inches.

b. Count the number of eggs in a nest.

c. Measure the weight in kg of bear cubs.

d. Count the number of defective light bulbs in a case of bulbs.

Random Variable

A random variable, x, represents a quantity being measured.

Example B

Random Variables

a. x = amount of rain each day in inches.

b. x = the number of eggs in a nest.

c. x = the weight in kg of bear cubs.

d. x = the number of defective light bulbs in a case.

Discrete and Continuous Random Variables

i. When a random variable x can take on only countable values

(such as 0, 1, 2, 3, . . .), then x is said to be a Discrete Random

Variable.

ii. When a random variable x can take on any value in an interval,

then x is said to be a Continuous Random Variable.

Example C

Discrete vs Continuous Random Variables

Which of the random variables in Example A, parts a-d, are discrete

and which are continuous?

Random Variables and Probability Distributions (Page 2 of 23)

Example D

Discrete vs Continuous Random Variables

Which of the following random variables are discrete and which are

continuous?

a. The number of students in a section of a statistics course.

b. The air pressure in an automobile tire.

c. The number of osprey chicks living in a nest.

d. The height of students at Palomar.

e. The mpg of randomly selected vehicles on a highway.

f. The time it takes a student to register for spring semester.

Probability Distribution

A probability distribution is an assignment of

probabilities to specific values of a random variable

(discrete) or to a range of values of a random

variable (continuous). A probability distribution is

basically a relative frequency distribution organized

in a table. Recall: The sum of all probabilities must

be one.

Mean and

Standard

Deviation of a

Discrete

Probability

Distribution

Roll a Die

P(x)

x

1

1/6

2

1/6

3

1/6

4

1/6

5

1/6

6

1/6

For a discrete random variable x and probability of

that variable, P(x):

mean = expected value = ? = " x ! P(x)

standard deviation = # =

When the random variable is

given as ranges of numbers,

set x equal to the midpoint of

each range. See Table.

Age

Range

18-24 years

25-34 years

" (x $ ? )

2

! P(x)

Midpoint of

the Range, x

21 years

29.5 years

P(x)

.26

.34

Random Variables and Probability Distributions (Page 3 of 23)

Example 1

Dr. Fidget developed a test

to measure boredom

tolerance. He administered

it to a group of 20,000

adults. The possible scores

were 0, 1, 2, 3, 4, 5, and 6,

with 6 indicating the highest

tolerance for boredom. The

results are shown.

Score

x

0

1

2

3

4

5

6

Number

(frequency, f )

1400

2600

3600

6000

4400

1600

400

Probability

P(x) = f / 20000

a. Find the probability (relative frequency) of each score and

construct a probability distribution in the table above. Let L1 be

the scores, x, L2 be the frequency, and L3 be the probabilities, P(x).

On the TI-83: L 2 / 20,000 ! L 3 .

b. Graph the probability

distribution as a

histogram of

probability versus test

score. What is the total

area of the bars?

c. Compute the expected value of the test scores and the standard

deviation. TI-83: 1-Var Stats Lx, LP(x). Use x = ? , and ! x = ! .

d. Topnotch Clothing Co. wants to hire someone with a score of 5 or

6 to operate is machinery. What is the probability that a randomly

selected person has a score of 5 or 6?

Random Variables and Probability Distributions (Page 4 of 23)

Exercise 7

Data was collected over 208 nights tabulating the number of room

calls in a night requiring a nurse.

a. Use the relative frequency to find P(x). In words, what does each

P(x) represent?

x

f

P(x)

36

6

37

10

38

11

39

20

40

26

41

32

42

34

43

28

44

25

45

16

b. Graph the probability distribution. Completely annotate the graph.

P(x)

x = number of room calls per night requiring a nurse

c. Estimate the probability that on a randomly selected night there

will be between 39 and 43 (inclusive) room calls requiring a

nurse.

d. Find the expected number of room calls requiring a nurse. Find

the standard deviation of the distribution.

Random Variables and Probability Distributions (Page 5 of 23)

Exercise 8

In 1851 the percent age distribution of nurses (to the nearest year) in

Great Britain was:

Age 20-29 30-39 40-49 50-59 60-69 70-79 80+

24.5 34.5 44.5 54.5 64.5 74.5 84.5

x

5.7

9.7

19.5 29.2

25

9.1 1.8

%

a. Use a histogram to graph the probability distribution. Completely

annotate the graph.



b. Find the probability that a randomly selected British nurse in 1851

would be 60 years or older.

c. What is the expected value (the ¡°balance point¡± on the graph) and

standard deviation of the age of a British nurse in 1851?

Ignore exercises 15-17 in section 5.1.

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