Prob. Dist. FunctionsDiscreteFamous -1

• The Binomial distribution

• The Poisson distribution

Prob. Dist. FunctionsContinuousFamous -2

• The Standard Normal distribution

• The Normal distribution

Agenda

The Binomial distribution

Status 1 with

probability k

Status 2 with

probability 1-k

N cases

X presents the number of cases of status i

Fail

Defected

Correct

Success

not defected

False

Ex:

The Binomial distribution

Status 1 with

probability k

Status 2 with

probability 1-k

X presents the number of cases of status i

If X represents the number of

cases of status 1 then p = k

Else

If X represents the number of

cases of status 2 then p = 1-k

A multiple choice test contains 25 questions,

each with four answers. Assume a student just

guesses on each question.

(a) What is the probability that the student

answers at least 20 questions correctly?

(b) What is the probability the student answers

less than 5 questions correctly?

Because not all airline passengers show up for

their reserved seat, an airline sells 125 tickets for

a flight that holds only 120 passengers. The

probability that a passenger does not show up is

0.10, and the passengers behave independently.

(a) What is the probability that every passenger

who shows up can take the flight?

(b) What is the probability that the flight departs

with empty seats?

Prob. 1

Prob. 9

Prob. 3

Prob. 4

Prob. 5

Prob. 6

Prob. 7

Prob. 8

Prob. 2

The Poisson distributionConsider the following examples:

1. flaws in rolls of textiles

2. calls to a telephone exchange

3. Car accidents on the highway.

the Poisson distribution describes the occurrence of a specific issue

(event) per an interval of (time, space, or volume) in the term of a rate ()

Interval of time or space or volume

Event

If the above example represents the arrival of 3 calls per 5

minutes then the rate () = 3

The Poisson distribution

Prob. 10

Prob. 11

Prob. 12

Prob. 13

Prob. 14

Prob. 15

The Poisson Distribution as a Limiting Form of the Binomial

In the case of the binomial, if n is large and p is close to 0, the Poisson distribution

can be used, with = np, to approximate binomial probabilities. If p is close to 1,

we can still use the Poisson distribution to approximate binomial probabilities by

interchanging what we have defined to be a success and a failure, thereby

changing p to a value close to 0.

Ex: In a certain industrial facility accidents occur infrequently. It is known that

the probability of an accident on any given day is 0.005 and accidents are

independent of each other.

(a) What, is the probability that in any given period of 400 days there will be

an accident on one day?

(b) What is the probability that there are at most three days with an accident?

Solution: Let. X be a binomial random variable with n = 400 and p = 0.005.

Thus = np = 2.

Using the Poisson approximation,

(a) P(X = 1) =

(b) P(X < 3) =

The Normal dist.

The Standard-Normal dist.

From Normal to Standard-Normal dist.

(a) z = (30 − 24)/3.8 = 1.58; P(X > 30) = P(Z > 1.58) = 0.0571.

(b) z = (15 − 24)/3.8 = −2.37; P(X > 15) = P(Z > −2.37) = 0.9911. He is late

99.11% of the time.

(c) z = (25 − 24)/3.8 = 0.26; P(X > 25) = P(Z > 0.26) = 0.3974.

(d) z = 1.04, x = (3.8)(1.04) + 24 = 27.952 minutes.

(e) Using the binomial distribution with p = 0.0571, we get

b(2; 3, 0.0571) =

Ex: A lawyer commutes daily from his suburban home to his midtown office.

The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8

minutes. Assume the distribution of trip times to be normally distributed.

(a) What is the probability that a trip will take at least 1/2 hour?

(b) If the office opens at 9:00 A.M. and he leaves his house at 8:45 A.M. daily, what

percentage of the time is he late for work?

(c) If he leaves the house at 8:35 A.M. and coffee is served at the office from 8:50

A.M. until 9:00 A.M., what is the probability that he misses coffee?

(d) Find the length of time above which we find the slowest 15% of the trips.

(e) Find the probability that 2 of the next 3 trips will take at least 1/2 hour.

The finished inside diameter of a piston ring

is normally distributed with a mean of 10 centimeters

and a standard deviation of 0.03 centimeter.

(a) What proportion of rings will have inside diameters

exceeding 10.075 centimeters?

(b) What is the probability that a piston ring will have

an inside diameter between 9.97 and 10.03

centimeters?

(c) Below what value of inside diameter will 15% of the

piston rings fall?

The heights of 1000 students are normally

distributed with a mean of 174.5 centimeters and a

standard deviation of 6.9 centimeters. Assuming

that the heights are recorded to the nearest half-

centimeter, how many of these students would you

expect to have heights

(a) less than 160.0 centimeters?

(b) between 171.5 and 182.0 centimeters

inclusive?

(c) equal to 175.0 centimeters?

(d) greater than or equal to 188.0 centimeters?

In the November 1990 issue of Chemical

Engineering

Progress, a study discussed the percent purity

of oxygen from a certain supplier. Assume that the

mean was 99.61 with a standard deviation of 0.08.

Assume that the distribution of percent purity was

approximately normal.

(a) What percentage of the purity values would you

expect to be between 99.5 and 99.7?

(b) What purity value would you expect to exceed

exactly 5% of the population?

A soft-drink machine is regulated so

that it discharges an average of 200 milliliters per

cup. If the amount of drink is normally distributed

with a standard deviation equal to 15 milliliters,

(a) what fraction of the cups will contain more than

224 milliliters?

(b) what is the probability that a cup contains

between 91 and 209 milliliters?

(c) how many cups will probably overflow if 230

milliliter cups are used for the next 1000 drinks?

(d) below what value do we get the smallest 25% of

the drinks?

Prob. 16

Prob. 17

Prob. 18

Prob. 19