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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 = 1 μ . Therefore, m = 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:

Exponential graph with increments of 2 from 0-20 on the x-axis of μ = 4 and increments of 0.05 from 0.05-0.25 on the y-axis of m = 0.25. The curved line begins at the top at point (0, 0.25) and curves down to point (20, 0). The x-axis is equal to a continuous random variable.

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 .

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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 ;

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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

Exponential graph with the curved line beginning at point (0, 0.25) and curves down towards point (∞, 0). Two vertical upward lines extend from points 4 and 5 to the curved line. The probability is in the area between points 4 and 5.

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).

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

b. Find the 50 th percentile.

Exponential graph with the curved line beginning at point (0, 0.25) and curves down towards point (∞, 0). A vertical upward line extends from point k to the curved line. The probability area from 0-k is equal to 0.50.

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 = l n ( 0.50 ) - 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 = l n ( 1 A r e a T o T h e L e f t ) 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.

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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.

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Questions & Answers

conceptual approach to limits
lameck Reply
how are limits derived?
lameck
an entire section of calculus is devoted to that explanation.
Pitior
what is statistics?
Martin Reply
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
Maham Reply
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
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Maham
yes use pnorm in r
Venkat
pnorm(2.3,mean=0,sd=1)
Venkat
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Maham
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Venkat
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Maham
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Venkat
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Venkat
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Maham
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Venkat
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Maham
ti83
Venkat
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Alwin
descriptive stats
Venkat
inferential stats
Venkat
outlier treatment
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Alwin
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Venkat
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Venkat
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Venkat
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Alwin
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Venkat
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Venkat
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Jameel
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Jameel
wer is the problem
Venkat
how find straight line equation in regression
Maham Reply
u can find using excel
Venkat
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Venkat
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Venkat
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Venkat
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Venkat
by giving a value to x,y
Ibrokhim
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Maham
x:1,2,3,4,5 y:2,5,6,8,9
Maham
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Venkat
y=0.9+1.7x
Venkat
reg eq is y=0.9+1.7x
Venkat
slope = 1.7
Venkat
yintercept = 0.9
Venkat
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Maham
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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
sodiq Reply
what is statistics
Bhavani Reply
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
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Madanapalle
no easy way
Pitior
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Pitior
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Bhavani
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Bhavani
Threads*
Bhavani
Finding correlation and regression
Langat Reply
explain statistics whether it is a science or arts or both
ratnapriya Reply
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?
Peachy Reply
inferential statistics is what?
Seyi Reply
in which we make infrences (hypothsis)
asad
surpose a data set of 2,3,5,6,1,4 are given find median
lucy Reply
fastest finger please
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
lucy Reply
my body pls
lucy
67
Benmike
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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)
Chinwendu Reply
how to find median
Hrishe Reply
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
asad
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?
thomas Reply
Any clues
Virtual

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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