# 5.3 The exponential distribution

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The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.

Let X = amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes.

X is a continuous random variable since time is measured. It is given that μ = 4 minutes. To do any calculations, you must know m , the decay parameter.

$m=\frac{1}{\mu }$ . Therefore, $m=\frac{1}{4}=0.25.$

The standard deviation, σ , is the same as the mean. μ = σ

The distribution notation is X ~ Exp ( m ). Therefore, X ~ Exp (0.25).

The probability density function is f ( x ) = me - mx . The number e = 2.71828182846... It is a number that is used often in mathematics. Scientific calculators have the key " e x ." If you enter one for x , the calculator will display the value e .

The curve is:

f ( x ) = 0.25 e –0.25 x where x is at least zero and m = 0.25.

For example, f (5) = 0.25 e −(0.25)(5) = 0.072. The postal clerk spends five minutes with the customers.

The graph is as follows:

Notice the graph is a declining curve. When x = 0,

f ( x ) = 0.25 e (−0.25)(0) = (0.25)(1) = 0.25 = m . The maximum value on the y -axis is m .

## Try it

The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes. Write the distribution, state the probability density function, and graph the distribution.

X ~ Exp (0.125); f ( x ) = 0.125e –0.125 x ;

a. Using the information in [link] , find the probability that a clerk spends four to five minutes with a randomly selected customer.

a. Find P (4< x <5).
The cumulative distribution function (CDF) gives the area to the left.
P ( x < x ) = 1 – e –mx
P ( x <5) = 1 – e (–0.25)(5) = 0.7135 and P ( x <4) = 1 – e (–0.25)(4) = 0.6321

## Note

You can do these calculations easily on a calculator.

The probability that a postal clerk spends four to five minutes with a randomly selected customer is P (4< x <5) = P ( x <5) – P ( x <4) = 0.7135 − 0.6321 = 0.0814.

On the home screen, enter (1 – e^(–0.25*5))–(1–e^(–0.25*4)) or enter e^(–0.25*4) – e^(–0.25*5).

b. Half of all customers are finished within how long? (Find the 50 th percentile)

b. Find the 50 th percentile.

P ( x < k ) = 0.50, k = 2.8 minutes (calculator or computer)

Half of all customers are finished within 2.8 minutes.

You can also do the calculation as follows:

P ( x < k ) = 0.50 and P ( x < k ) = 1 – e –0.25 k

Therefore, 0.50 = 1 − e −0.25 k and e −0.25 k = 1 − 0.50 = 0.5

Take natural logs: ln ( e –0.25 k ) = ln (0.50). So, –0.25 k = ln (0.50)

Solve for k : $k=\frac{ln\left(0.50\right)}{-0.25}=2.8$ minutes. The calculator simplifies the calculation for percentile k . See the following two notes.

## Note

A formula for the percentile k is $k=\frac{ln\left(1-AreaToTheLeft\right)}{-m}$ where ln is the natural log.

## Collaborative exercise

On the home screen, enter ln(1 – 0.50)/–0.25. Press the (-) for the negative.

c. Which is larger, the mean or the median?

c. From part b, the median or 50 th percentile is 2.8 minutes. The theoretical mean is four minutes. The mean is larger.

#### Questions & Answers

conceptual approach to limits
how are limits derived?
lameck
an entire section of calculus is devoted to that explanation.
Pitior
what is statistics?
statistics :- can be defined as the branches of mathematics that deals with the summarizing, analysing,organization and interpretation of data.
Usman
well said
Venkat
can we find Z value on calculator with out using Z table
no
Pitior
why
Maham
can another way is possible ?
Maham
Well you could make a table. And as the function you use the one used at the z table
Luca
The normal function is only one way, so you can only try using different numbers until you get the probability that you have. So that is easier if you have a table
Luca
me don't know nothing about z table and don't know how to see the z value on table can you tell me please how see the value on table
Maham
The z table is the table of the standard normal distribution
Luca
You can look it up on internet, its easier than writing down the normal distribution function (with an integral) and doing a table in the calculator
Luca
OK thanks luca
Maham
yes use pnorm in r
Venkat
pnorm(2.3,mean=0,sd=1)
Venkat
pnorm?
Maham
do u have r software
Venkat
no
Maham
its with tht u will get
Venkat
or type in google
Venkat
z mathportal calculator
Venkat
calculator
Venkat
OK venkat thanks
Maham
welcome
Venkat
have calculator but don't know how find z value
Maham
ti83
Venkat
hey guys I'm from computer background so what are the concepts I supposed to prepare for interview in statistics
Alwin
descriptive stats
Venkat
inferential stats
Venkat
outlier treatment
Venkat
boxplot
Venkat
ok
Alwin
assumption of linear regression
Venkat
logistic regression
Venkat
k means clustering
Venkat
exact syllabus?
Alwin
type. analytics vidya interview questions statistics
Venkat
listen. data also
Venkat
like this forum
Jameel
My question is "is it only stats?"
Jameel
wer is the problem
Venkat
how find straight line equation in regression
u can find using excel
Venkat
or r studio
Venkat
for regression
Venkat
shall i help
Venkat
im an expert
Venkat
by giving a value to x,y
Ibrokhim
first provide data
Venkat
ill solve and guve
Venkat
ive
Venkat
Maham
maham you posted data
Venkat
Venkat
ok
Maham
x:1,2,3,4,5 y:2,5,6,8,9
Maham
regredsion equation is
Venkat
y=0.9+1.7x
Venkat
reg eq is y=0.9+1.7x
Venkat
slope = 1.7
Venkat
yintercept = 0.9
Venkat
Venkat
thanx venkat naveen😊
Maham
welcome
Venkat
the tenth percentile for land selling at jabi is 35,000 and the nineteenth percentile for the land price in the same area is 225,what is the 10_90 percentile range
what is statistics
statistics is the beach of mathematics which deals with collection ,organisation, presentation, analysis and interpretation of numerical data
Saeed
oh but interpretation of data, like what and how? 🤔
Bhavani
interpretation: Think in a way that you have given a company year turnover and you have a record of 100years and data set is like (Year,Turnover). Now with that data you can interpret many thing how was the company growth, when were the losses and other things
Akash
interpretation: it is a process in which we make a decision about a population on the basis of sample data . example: if we want to interpret the average income of employees for upcoming year so we have to interpret the income of employees on the basis of previous year's income of those employees
Saeed
thank you saeed, Akash. I understood.
Bhavani
how to remember all this formulas easy ly
no easy way
Pitior
best way is to do as many problems as possible
Pitior
Oh
Bhavani
is this the only one room? or separate room for separate users? 🤔
Bhavani
Bhavani
Finding correlation and regression
explain statistics whether it is a science or arts or both
I would say art is a creation. A chef is an artist. They create new dishes just like the painters. I believe one who creates something new, is an artist. So, Statistics is also an art, if you know it, you can create some new formula, theory, law, etcetera. It is also Science. So yes, it is both.
Rohan
how do you use the normal distribution table when testing the hypothesis
Davia
I am sorry Davia, I cannot help you with that.
Rohan
percentages of all the possible outcomes are measured. This is so simple and bases on the questionnaire or interview schedule. It's just measuring the probability chances of high %age of the either part of the hypothesis ... dependent ..independent. data is classified on the basis of respondents
saifuddin
how find CV(x) and CV(y) if X: 3,7,5,4,6,9 &Y:4,9,6,4,7,8 please tell
Maham
what percent of the students would be expected to score above 95?
inferential statistics is what?
in which we make infrences (hypothsis)
surpose a data set of 2,3,5,6,1,4 are given find median
lucy
Mean (average) 4... Median (middle term) 3.5.. Mode (frequency) every element in a set has 1 frequrncy
Akash
i arrange the data set in ascending order. that is, 1,2,3,4,5,6. then find the data set that falls in the middle. in this case, 3 & 4 fall in the middle. you then sum and obtain the average. that is, (3+4)/2=3.5. therefore, 3.5 is the median.
Gbenga
both of you are correct.
Joseph
hello guys
Abasikponke
thanks
lucy
great to be here
King
how does a line graph look
King
hi
Davia
hello
lucy
pls who knows how line graph look like
King
line graph usually have a straight line running through axis
Dike
am new here anyone willing to orient me?
Timothy
find the media of the following numbers 61,64,67,70,73
my body pls
lucy
67
Benmike
what happen to your body@hana
Benmike
what is the percentile for the set of data in the class C and frequency F(c,f)given by (9.3-9.7,2) (9.8-10.2,5) (10.3-10.7,12) (10.8-11.2,17) (11.3-11.7,14) (11.8-12.2,6) (12.3-12.7,3) (12.8-13.2,1)
how to find median
arrange ascending and desending order than the mid value is Median
rajendra
ok
Hrishe
what if it is a group data
Oloyede
mean/ medium/ mode
Michelle
n\2 and n+1\2
An operational manager at a manufacturing company is interested in the level of satisfaction of computer buyers. The manager has developed a satisfaction scale of 1-10 to mark their level of understanding with the company.What is the population of the interest?
Any clues
Virtual