Probability Distributions

Text Text Text Text Text Text

Title

Random Variables and

Probability Distributions

Recall that one of the objectives of statistics is to make generalizations regarding

a specified population. And these generalizations are always subject to uncertainties due

to the limited information that can be obtained from sample observations. One of the

ways to deal with this problem is through the study of probability theories. Probability

theory provides a way to construct a model that theoretically describes the behavior of a

population that is associated with the statistical experiment involved.

5.1 Random Variable We recall that a statistical experiment is yields random outcomes. And there are

instances that we are just interested some of the details of the outcomes. For instance an

experiment of tossing a fair die twice, there would be 36 possible outcomes. If we are just

interested in the number of heads in the outcome of the toss, then we are only considering

once characteristic of the outcome of the experiment. And since the outcomes can vary

from sample to sample we may consider this characteristic our variable. Thus we define

what we mean by a random variable. A random variable is defined to be a function

whose value is a real number determined by each element in the sample space is called a

random variable. A random variable is usually denoted by a capital letter and specific

values of the random variable are represented by a small letter.

Example

For instance, in an experiment of tossing a coin thrice and we are only concerned

with the outcome of the number of heads occurring in the experiment we may associate

the number 0, 1, 2, and 3 to the number of head that may occur in a particular outcome.

To represent these values we may want to use a variable, a random variable. Random

since we are not definite about the values of our variable. We just know the possible

values it may take.

If we let X be the random variable that represents the number of tails in the

outcome then we have the following:

Given sample space S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

RECALL:

The sample space of a given experiment is the set of all possible outcomes. And

so if we define our random variable based on that sample space we can categorize a

random variable in the following manner: Discrete and Continuous.

First we define the following:

Discrete Sample Space - If a sample space contains a finite number of

possibilities or an unending sequence with as many elements as there whole numbers, it

is called a discrete sample space.

Continuous Sample Space – If a sample space contains an infinite number of

possibilities equal to a number of points on a line segment, it is called a continuous

sample space.

Example

Classify the following random variables as discrete or continuous.

a) the number of automobile accidents each year in Virginia discrete

b) the length of time to play 18 hole of golf continuous

c) the amount of milk produced by a certain cow per month continuous

d) the number of eggs laid each month by a specific hen discrete

e) the weight of grain in pound produced per acre continuous

Sample points TTT HTT, THT, TTH HHT, HTH, THH HHH

x 0 1 2 3

Types of random variable

Discrete Random Variable is a random variable which is defined on a discrete sample

space while a Continuous Random Variable is a random variable defined on a continuous

sample space.

Discrete Probability Distribution – A table or a formula listing all possible values that a

discrete random variable can assume, along with the associated probabilities, is called a

discrete probabilities distribution.

Example

1. In an experiment of tossing a coin three times the following sample space is obtained:

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. We define the random variable

X, as the number of head in an outcome. We summarize the result of the experiment and

identify the values of our random variable as well as the associated probability with each

value of the random variable.

2. Find the probability distribution given a random variable x defined as the sum of

the numbers when a pair of dice is tossed.

The following table illustrates all possible outcomes when a pair of dice is tossed and the

associated probability distribution for the

Outcomes for the Sum of Two Dice

x x x x x x

1, 1 2 2, 1 3 3, 1 4 4, 1 5 5, 1 6 6, 1 7

1, 2 3 2, 2 4 3, 2 5 4, 2 6 5, 2 7 6, 2 8

1, 3 4 2, 3 5 3, 3 6 4, 3 7 5, 3 8 6, 3 9

1, 4 5 2, 4 6 3, 4 7 4, 4 8 5, 4 9 6, 4 10

1, 5 6 2, 5 7 3, 5 8 4, 5 9 5, 5 10 6, 5 11

1, 6 7 2, 6 8 3, 6 9 4, 6 10 5, 6 11 6, 6 12

Usually it is convenient to represent all the probabilities associated with each

value of a random variable by a formula. And such formulas are called probability

functions or probability distributions. Thus we define the following terms:

Sample points TTT HTT, THT, TTH HHT, HTH, THH HHH

x 0 1 2 3

)x(f 8

1

8

3

8

3

8

1

x 2 3 4 5 6 7 8 9 10 11 12

)x(f 36

1

36

2

36

3

36

4

36

5

36

6

36

5

36

4

36

3

36

2

36

1

Probability Density Function – The function with values f(x) is called a probability

density function for a continuous random variable x if the total area under its curve and

above the x-axis is equal to 1 and if the area under the curve between any two ordinates a

and b gives the probability that the random variable x between a and b.

Mean and Variance of a Discrete Random Variable.

Given a random variable x with a probability distribution xf

x 1x 2x 3x … … … nx

)x(f )x(f 1 )x(f 2 )x(f 3 … … … )x(f n

The mean or expected value of the random variable x is given by

n

iii xfxxE

1

And the variance of x is given by

n

iii xfxXE

1

222

Example 1. Find the mean and variance of H, where H is a random variable which

represents the number of automobiles that are used for official business purpose on any

given workday by a certain company. The probability distribution for H is as follows:

H 1 2 3

F(H) = P(H=h) 0.3 0.4 0.3

Solution

23034023011

...xfxHEn

iii

n

iii hfhHE

1

222 60302340223021222

....

F(x)

G(x)

2. A shipment of 7 television sets contains 2 defectives were delivered by Tan

Electronic Company at a certain mall in Manila. If Dusit Hotel makes a random purchase

of 3 of the sets. If G is the number of defective sets purchased by the hotel, find the mean

and variance of G.

Solution

First we have to determine the probabilities associated with each value of the random

variable G.

g 0 1 2

gf

We now have to find the probabilities where the g = 0, 1, and 2.

0f is the probability that the hotel purchases 3 television sets where none of the set is

defective. 1f is the probability that the hotel purchases 3 television sets where one of

the set is defective. And so on

Using our notions of finding probabilities we have the following computation:

N

ngf , where n is the number of outcomes in the event where the hotel purchases g

defective television sets and N is the total number of ways of selecting 3 television sets

out of the 7 sets that were shipped.

So we have,

35

337

737

!!

!CN

If g = 0,

Since in this event we are selecting none of the defective and 3 out of the 5 television sets

that are not defective we have the following computation:

10110

002

2

335

50235

*

!!

!*

!!

!C*Cn

Thus 35

100 f

If g = 1,

Since in this event we are selecting one of the defective and 2 out of the 5 television sets

that are not defective we have the following computation:

20210

112

2

225

51225

*

!!

!*

!!

!C*Cn

Thus 35

201 f

If g = 2,

Since in this event we are selecting two of the defective and one out of the 5 television

sets that are not defective we have the following computation:

515

222

2

115

52215

*

!!

!*

!!

!C*Cn

Thus 35

52 f

Completing our probability distribution we have the following table.

g 0 1 2

gf 10/35 20/35 5/35

Computing for the mean and variance of the random variable G,

7

6

35

30

35

52

35

201

35

100

1

n

iii gfggE

50.40816326

gfg

GE

n

iii

35

5

7

62

35

20

7

61

35

10

7

60

222

1

2

22

Exercises

1. In an experiment of selecting 3 persons to form a committee from a set of 4 boys and 3

girls. Let H represent the number of boys on the committee.

2. Find the number of expected Jazz records when 4 records are selected at random from

a collection consisting of 5 jazz records, 2 classical records, and 3 polka records.

3. A coin is tossed three times. Let Y be the random variable that represents the number

of tails. Find the probability distribution of Y. Find the mean and variance of the

probability distribution of the random variable Y.

4. In an experiment of tossing a dice first and then tossing a coin, where the coin is tossed

once if the dice resulted in an even number and twice if the dice resulted to an odd

number. Find the probability distribution of the random variable Y, where Y represents

the number or heads in the outcome.

5. Let R be a random variable with the following probabilities

r 0 1 2 3 4

rf 1/12 3/12 5/12 2/12 1/12

Find the mean and variance of R.

6. The probability distribution of a discrete random variable X is given by the following

xx

xxf

4

5

4

5

14 for 43210 ,,,,x

Find the mean and variance of X

7. A die is thrown twice and the number of times an odd number comes up is recorded.

a. Construct the probability distribution table for the random variable X, the number

of times an odd number comes up.

b. Find the expected value and variance of X.

5.2 SOME DISCRETE PROBABILITY DISTRIBUTIONS

Probability distributions describe the behavior of our random variable and this is

presented either in a tabular form like the probability histogram or in a tabular form. And

often we just need to generalize and summarize how to describe the distribution of the

random variable. This is obtained by representing the probability distribution by means of

a mathematical function or formula. And in practice, we only need a handful of important

discrete probability distributions that would describe most random variable that can be

encountered in real world applications.

Some of the discrete probability distributions are the following: Binomial

distribution, Hypergeometric distribution, and Poisson distribution

Binomial Distribution

A binomial experiment has the following properties:

The experiment consists of n repeated trials

Each trial results in an outcome that may be classified as a success or a failure

The probability of a success, denoted by p, remains constant from trial to trial.

The repeated trials are independent.

Usually if the first 3 conditions are already met, the last condition is presumably a

forgone conclusion. For a random variable X to have a binomial distribution, the

conditions of a binomial experiment must be satisfied.

The number x of success of a random variable X in n trials of a binomials experiment is

called a binomial random variable.

If a binomial experiment can result in a success with probability p and the failure

with the probability q = 1- p, then the probability distribution of the binomial random

variable X, the number of success in n independent trials is

xnxqpx

np;n;xb

for n,...,,,,x 3210

Note: !x!xn

!nC

x

nxn

The mean and variance of the binomial distribution p;n;xb are given by the formulas

np and npq2

Example

1. Find the probability of obtaining exactly three 2’s if an ordinary di ce is tossed 5 times.

Solution:

Suppose X is the random variable representing the number of 2’s occurring in tossing a

dice 5 times.

Check if the conditions of the binomial experiment are satisfied.


There are 5 repeated trials of tossing a dice


The outcome can be classified as a success when the result of the dice is 2 and a

failure if the outcome is not 2.


The probability of a success on each of the 5 trials is 6

1 and the probability of failure

is6

5.


We conclude that the trials are independent from one another since the result of the

first toss does not affect of the resul t of the next toss.

Thus we have, n = 5, 6

1q ,

6

5q and 3x .

03206

5

6

1

3

5

6

153

353

.qpx

np,n;xb xnx

2. A survey in Cavite indicated that nine out of ten cars carry automobile liability

insurance. If 4 cars in Cavite are involved in accidents, what is the probability that:

Solution If we consider the random variable X to be present the number of automobiles carrying

liability insurance out of the 4 cars involved in an accident. Checking if the conditions of

the binomial experiment are satisfied, we have the following


Image taken from the book “The Cartoon Guide to Statistics by Larry

Cognick and Woollcott Smith”

The repeated trial can than can be considered is the checking of the automobile if it

has a liability insurance. Thus, there are 4 repeated inspections whether the 4 accidents of

automobiles carries with them a liability insurance.


The inspection can result to a success if the automobile associated in the accident

carries a liability insurance otherwise the result is considered a failure.


The probability of a success on each of the 4 trials is 10

9 and the probability of

failure is10

1.


We conclude that the trials are independent from one another since the result of the

first inspection does not affect of the result of the next inspection.

Based from the information given, we have, n = 4, 10

1q ,

10

9p .

a) No more than two of the four drivers have liability insurance?

What we want to find out is the following probability

2102 xPxPxPxP

Using the formula for the binomial probabili ty we have the following computations,

0001010

1

10

9

0

4

10

940

40

.qpx

np,n;xb xnx

0036010

1

10

9

1

4

10

941

31

.qpx

np,n;xb xnx

0486010

1

10

9

2

4

10

942

22

.qpx

np,n;xb xnx

Thus, 2102 xPxPxPxP = 0.0001 + 0.0036 + 0.0486 = 0.0523

b) Exactly 3 have liability insurance?

2916010

1

10

9

3

4

10

9433

13

.qpx

np,n;xbxP xnx

c) All of the cards involved in the accidents carries a liability insurance.

6561010

1

10

9

4

4

10

9444

04

.qpx

np,n;xbxP xnx

Hypergeometric Distribution

A Hypergeometric experiment has the following properties:

A random sample of size n is selected from a population of N items.

k of the N items may be classified as success and N – k as failures.

And a random variable defined as the number of successes in a Hypergeometric

experiment is called a Hypergeometric Random Variable.

If the population of size contains k items labeled as “success” and N – k items

labeled as “failures” then the probability distribution of the Hypergeometric random

variable X, the number of successes in a random sample of size n, is

n

N

xn

kN

x

k

k,n,N;xh for n,...,,,x 3210

The mean and variance of the Hypergeometric distribution k,n,N;xh are given by the

formulas

N

nk and

N

kN

N

nk

N

nN

1

2

Example

1. If 5 cards are dealt from a standard deck of 52 playing cards what is the

probability that 3 will be hearts?

Solution:

Clearly we can label all heart cards as our success. Hence, k = 13, since there are 13 heart

cards. And since we are selecting 5 cards from the deck our sample size n = 5. Thus we

have the following:

N k

n

x

08150

5

52

2

39

3

13

135523 .

n

N

xn

kN

x

k

k,n,N;xh

2. If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the

probability that

a) Exactly 2 of them will be face cards?

b) At least 1 of them will be a queen?

Poisson Distribution

A poisson experiment has the following properties:

The number of outcomes occurring in one time interval or a specified region is

independent of the number of outcomes that occur in any other disjoint time interval or

region space.

The probability that a single outcome will occur during a very short time interval

or in a small region is proportional to the length of time interval or the size of the region

and does not depend on the outcomes occurring outside this time interval or region.

The probability that more than one outcome will occur in a very short time

interval or a small region is very small and can be assumed to be negligible.

The number X of success in a poisson experiment is called a poisson random variable.

The probability distribution of a poisson random variable X representing the

number of outcomes occurring in the given time interval or specified region is

!x

e;xp

x

for ,...,,,x 3210

Where is the average number of outcomes occurring in the given time interval or

specified region and ....e 718282

Example

1. The average number of days school is closed due to floods during the rainy season

in a city in Pampanga is 4. What is the probability that the schools in this particular city

in Pampanga will close for 6 days during a rainy season?

Solution:

104206

446

64

.!

e;xp

2. The average number of dagang bukid per acre in a 5-acre rice field in Baguio is

estimated to be 10. Find the probability that a given acre contains more than 3 dagang

bukid.

Solution:

To find the probability that a given acre contains more than 3 dagang bukid, we

need to find the probability of its complement, since it is easier to find. And just use the

theorem on probabilities for complementary events. Suppose X is our random variable

representing the number of dagang bukid in a 2 acre rice field in Baguio. Thus we have

the following: 313 xPxP

EXERCISES

1 On the average, the intersection of Taft Avenue and Buendia results in 3 traffic

accidents per month. What is the probability that in any given month at this intersection

a. Exactly 5 accidents will occur?

b. Less than 3 accidents will occur?

c. At least 2 accidents will occur?

2 A basketball player’s shooting average is 0.25, what is the probability that he gets

exactly 1 shoot in his next 5 times attempt to shoot the ball

3 A multiple-choice quiz has 10 questions, each with 4 possible answers of which

only one is correct. What is the probability that sheers guess work yields from 3 to 6

correct answers?

4 If probability that a patient recovers from a leukemia is 0.4. And if 15 people are

known to have contracted this disease, what is the probability that

b) At least 13 survive

c) From 3 to 5 person survive

d) Exactly 5 survive.

5 In a Metro Manila, MMDA says that the need for money to by drugs is given as

the reason for 55% of all thefts. What is the probability that exactly 2 of the next 4 theft

cases-reported to MMDA resulted from the need for money to buy drugs?

6 A homeowner plants 5 bulbs selected at random from a box containing 5 rose

bulbs and 4 sampaguita bulbs. What is the probability that he planted 2 sampaguita bulbs

and 3 rose bulbs?

7. A professor in biology gave a multiple choice quiz with 10 items, each with 5

possible answers and only one of which is correct.

a) What is the probability that a student took the test my merely guessing and got a score

of 5?

b) What is the probability that merely guessing the answers from the test would yield a

score of 4 to 8?

c) What is the probability that merely guessing the answers from the test would yield a

score of at least 5?

8. What is the probability that a waiter will refuse to serve alcoholic drinks to only 2

minors if he randomly checks the Identification cards of 5 students from among the 10

students where 4 of which are not of legal age?

9. The average number of patients arriving a t the emergency room of Philippine General

Hospital (PGH) on Monday nights between 9:00 pm up to 12:00 midnight is 5. If we

assume that the patients arrive at random and independently, what is the probability that

less than 5 patients arrive at the emergency room of PGH on a Monday night from 9:00

pm to 12:00 midnight?

10 . A box contains10 red marbles and 15 blue marbles and 5 marbles are selected at

random from the box.

a) What is the probability of obtaining at least 3 red marbles?

b) What is the probability of obtaining at most 2 blue marbles?

c) What is the probability of obtaining exactly 1 red marble?

11. Suppose that the average number of earthquakes experienced in Mindanao is 10 per

year. What is the probability that on a given year, Mindanao will experience at least 5

earthquakes?

12. In certain computer shop, the typist commits on the average two typographical error

per page. What is the probability that the typist makes

a) 3 or more errors

b) at least 1 error

c) no errors

13. In Davao, the probability that a household has a Pomelo tree in their backyard is 0.35.

Find the probability that 4 out of the 10 randomly selected houses has a Pomelo tree in

their backyard.

14. Batanes is hit by 8 storms per year on the average. What is the probabili ty that on a

certain year, Batanes will be hit by at least 5 storms?

15. Warranty records show that the probability that a new car needs repair in the first 90

days is 0.10. If a sample of ten new cars is selected,

a. what is the probability that none needs a warranty repair?

b. what is the probability that at least 3 needs a warranty repair?

c. what is the probability that from 5 to 8 (inclusive) needs a warranty repair?

d. what is the probability that at most 6 needs a warranty repair?

16. The quality control manager of Mandy's Cookies is inspecting a batch of chocolate

chip cookies that has just been baked. If the production process is in control, the average

number of chip parts per cookie is 6.0. What is the probability that in any particular

cookie being inspected,

a. exactly 5 chip parts will be found?

b. more than 3 chip parts will be found?

c. less than 7 chip parts will be found?

5.3 NORMAL DISTRIBUTION

The normal distribution is one of the most important continuous distribution in

the entire field of statistics. And the graph of this distribution is called the normal curve.

This distribution is sometimes called the Gaussian distribution in honor of Karl Friedrich

Gauss, who derived its equation.

Properties of the normal curve

It is a bell-shaped curve

The mode, which is the point on the horizontal axis where the curve is a

maximum, occurs at x .

The curve is symmetric about a vertical axis through the mean, .

The normal curve approaches the horizontal axis asymptotically as we proceed in

either direction away from the mean. (The graph approaches the x -axis but the

graph will never intersect the x-axis).

The total area under the curve and above the horizontal axis is equal to 1.

A continuous random variable X having the bell-shaped distribution is called a normal

random variable. The mathematical equation for the probability distribution of the

normal random variable depends on two parameters and ; its mean and standard

deviation. Thus we denote the probability density of X by ,;xN .

REMARK

It is difficult to compute for the probabilities of a normal random variable using

the above formula. However, another way of calculating such probabilities is through the

transformation of a normal random variable to its corresponding standard normal

random variable. By transforming a normal random variable to a standard normal

random variable we can now determine probabilities of the said random variable. Thus

we define the standard normal random variable and its distribution.

NOTE:

If X is a normal random variable with mean and variance 2 , then the

equation of the normal curve is

2

2

1

2

1,;

x

exN , for x , where

...71828.2 ...14159.3 eand

Standard Normal Distribution: The distribution of a normal random variable with

mean 0 and standard deviation 1 is called a standard normal distribution.

In order to transform a normal random variable to a standard normal one, we use

the following formula:

XZ

By using the table for the standard normal random variable, we can now determine the

probability of any normal random variable by transforming the given random variable to

its corresponding standard normal random variable.

EXAMPLES

1. Given a normally distributed random variable X with mean 18 and standard

deviation of 2.5, find

a) 15XP

Solution:

115102152

181515 ..ZP

.ZPXP

Referring to Appendix A.

b) 2117 XP

Solution:

54030

3446088490

4021

2140

52

1821

52

18172117

.

..

.ZP.ZP

.Z.P

.Z

.PXP

c) The value of k such that 25780.kXP

Solution:

To find the value of k, we use the formula for transforming the random variable X

to a standard normal random variable that is;

2578052

18.

.

kZPkXP

By referring to our standard normal table, we would find that the value of z is

650. such that the area under the curve or the probability is 0.2578. Thus,

65052

18.

.

k

which implies that 18161852650 ..*.k

2. Given a normal distribution with 50 and 10 , find the probability that X

assumes a value between 45 and 62.

SOLUTION:

We are asked to determine 6245 xP . From the table that we have the

following:

456262456245 xPxPxPxP

Transforming X to Z we have the following:

5010

5

10

50451 .

XZ

51

10

12

10

50622 .

XZ

Thus we have,

502121506245 .ZP.ZP.Z.PxP

57640

3085088490

.

..

Exercises

= -

1. Given a normally distributed random variable X with mean 18 and standard deviation

of 2.5, find the value of k such that 15390.kXP

2. A certain type of storage battery last on the average 3.0 years, with a standard

deviation of 0.5 years. Assuming that the battery lives are normally distributed, find the

probability that a given battery will last less than 2.3 years.

3. An electrical firm manufactures light bulbs that have a length of life that is normally

distributed with mean equal to 800 hours and a standard deviation of 40 hours. Find the

probability that a bulb burns between 778 and 834 hours.

4. If the average height of miniature poodles is 30 centimeters, with a standard deviation

of 4.1 cm, what percentage of miniature poodles exceeds 35 cm in height, assuming that

the height follows a normal distribution and can be measured to any desired degree of

accuracy?

5. The quality grade-point averages of 300 college freshmen follow approximately a

normal distribution with a mean of 2.1 and a standard deviation of 0.8. How many of

these freshmen would you expect to have a score between 2.5 and 3.5 inclusive if the

point averages are computed to the nearest tenth?

6. A set of final examination grades in an introductory statistics course was found to be

normally distributed, with a mean of 73 and a variance of 64.

a. What is the probability of getting a grade of 91 or less in this exam?

b. What percentage of students scored between 81 and 89?

c. Only 5% of the students taking the test scored higher than what grade?

7. Plastic bags used for packaging produce re manufactured so that the breaking strength

of the bag is normally distributed with a mean of 5 pounds per square inch and a standard

deviation of 1.5 pounds per square inch.

a. What proportion of the bags produced have a mean breaking strength of between 5

and 5.5 pounds per square inch?

b. What is the probability that a randomly selected bag will have a mean breaking

strength of at least 6 pounds per square inch?

c. What percentage of the bags have a mean breaking strength of less than 4.17

pound per square inch?

d. Between what two values symmetrically distributed around the mean will 95% of

the breaking strengths fall?

8. If we know that the length of time it takes a college student to find a parking spot in

the university parking lot follows a normal distribution with a mean of 3.5 minutes and a

standard deviation of 1 minute, find the probability that if we select 36 randomly

selected college students, the average time it would take for them to find a parking spot is

a) less than 3.2 minutes?

b) between 3.4 and 3.7 minutes?

c) more than 3.8 minutes?

Summary 1. A random variable is defined to be a function whose value is a real number determined

by each element in the sample space is called a random variable.

2. A Discrete Random Variable is a random variable which is defined on a discrete sample

space while a Continuous Random Variable is a random variable defined on a

continuous sample space.

3. Some of the discrete probability distributions are the following: Binomial distribution,

Hypergeometric distribution, and Poisson distribution.

4. Properties of the binomial experiment





5. The most widely used continuous distribution is the normal distribution. However

calculation of probabilities in this type of distribution is difficult to derive even with the

use of computers. For this reason i t is necessary to transform the random variable into a

standardized random variable, that is, standard normal random variable.

The central limit theorem explains why the normal distribution is the most widely

used distribution. It is applicable to the stock market fluctuations, students’ grades,

price of canned goods, weight of people in a city, amount of mercury in a river, thus

practically everywhere. For instance, the price of canned goods are influenced by the

price of gasoline, price of tin can used for packing, labor cost in producing the

goods, type of product to be placed in the canned good, location of the factory that

manufactures the canned goods, etc. These are all unrelated factors that influence

the price of the canned goods but when considered together, the effect you’ll get is a

normal distribution

Cartoon illustration taken from the book “Cartoon Guide to Statistics” by Larry Cognick and Woollcott

Smith

Facts and Figures in Statistics

Abraham De Moivre (1667 - 1754) was a French-born

mathematician who pioneered the development of analytic

geometry and the theory of probability.

Chapter Review Write the letter that corresponds to the correct answer.

For #’s 1-4, given the following probability distribution of a random variable x

x 0 1 2 3

f(x) 0.23 0.25 0.41 0.11

1. What is the probability that X is greater than 3?

a) 0.00 b) 0.11 c) 1.00 d) 0.25

2. What is the probability that X is even?

a) 0.00 b) 0.11 c) 1.00 d) 0.25

3. What is the probability that X is odd?

a) 0.00 b) 0.11 c) 1.00 d) 0.25

4. What is the mean of X

a) 0.00 b) 0.11 c) 1.00 d) 0.25

5. In an experiment where the probability of a success is 0.3, if you are interested in the

probability of 2 successes out of 5 trials, the correct probability is

a) 0.0774. b) 0.1600. c) 0.2613. d) 0.0016.

6. Which of the following does not describe a binomial experiment?

a) The number of trials is fixed.

b) There are exactly two possible outcomes for each trial.

c) The individual trials are dependent on each other.

d) The probability of failure is the same for each trial.

7. In a binomial experiment where n is the number of trials and p is the probability of

success, then the standard deviation for the resulting binomial distribution is given by

a) np b) pn 1 c) 1np d) pnp 1

8. The following distributions are discrete except the

a) Binomial distribution b) Hypergeometric distribution

c) Poisson distribution d) Normal distribution

For nos. 9 – 11, consider the following:

Given the following probability distribution:

X = x 0 1 2 3

P(X=x) 0.34 0.25 0.23

9. The value of P(X=2) is

a. 0.28 c. 0.15

b. 0.16 d. 0.18

10. The expected value of this probability distribution is

a. 0.5 b. 1.3 c. 1.8 d. 0.38

11. The variance of this probability distribution is

a. 1.35 c. 2.5

c. 1.16 d. 1.58

12. The following are all properties of the normal distribution EXCEPT

a. It is bell shaped

b. The total area under the curve is 1.

c. The mean is 0 and the standard deviation is 1.

d. It is symmetric about the mean µ.

13. P(Z<1.45) is equal to

a. 0.0735 c. 0.9265

b. 0.9192 d. 0.0808

14. P(Z>-0.65) is equal to

a. 0.7422 c. 0.7257

b. 0.2578 d. 0.2743

15. P(-1.03<Z<2.12) is equal to

a. 0.9830 c. 0.1515

b. 0.8485 d. 0.8315

16. There are 18 toys in a basket, of which 10 are cars and 8 are balls. A child randomly

picks 3 toys without replacement. Let X be a random variable, the number of balls

selected. What is the distribution of X?

a. Binomial c. Normal

b. Poisson d. Hypergeometric

17. The basketball player Shaq makes 45% of the free throws he tries. Find the

probability that in the next 4 throws, he will make exactly 3 hits?

a. 0.2 b. 0.3 c. 0.4 d. 0.5

18. In DLSU, there are 9 candidates from 2 political parties, 5 from TAPAT and 4 from

SANTUGON, aiming for 6 Student Council positions. Assuming that all candidates are

equally qualified for the positions, find the probability that 3 TAPAT candidates and 3

SANTUGON candidates will be elected for these positions.

a. 0.2865 b. 0.3589 c. 0.4768 d. 0.5556

19. An average of 0.8 accident per day occurs in a certain city. What is the probability

that no accident will occur in this city on given day?

a. 0.4493 b. 0.3980 c. 0.25 d. 0

20. Suppose that a random variable X has a normal distribution with mean 40 and

standard deviation 5. What is the probability that X is below 30?

a. 0.0228 a. 0.1587 c. 0.8413 d. 0.9772

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Probability Distributions