5.3 The exponential distribution  (Page 6/25)

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Chapter review

If X has an exponential distribution with mean μ , then the decay parameter is m = $\frac{1}{\mu }$ , and we write X Exp ( m ) where x ≥ 0 and m >0 . The probability density function of X is f ( x ) = me -mx (or equivalently $f\left(x\right)=\frac{1}{\mu }{e}^{-x/\mu }$ . The cumulative distribution function of X is P ( X x ) = 1 – e mx .

The exponential distribution has the memoryless property , which says that future probabilities do not depend on any past information. Mathematically, it says that P ( X > x + k | X > x ) = P ( X > k ).

If T represents the waiting time between events, and if T Exp ( λ ), then the number of events X per unit time follows the Poisson distribution with mean λ . The probability density function of P X is $\left(X=k\right)=\frac{{\lambda }^{k}{e}^{-k}}{k!}$ . This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf( λ , k ). The cumulative distribution function P ( X k ) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf( λ , k ).

Formula review

Exponential: X ~ Exp ( m ) where m = the decay parameter

• pdf: f ( x ) = me (– mx ) where x ≥ 0 and m >0
• cdf: P ( X x ) = 1 – e (– mx )
• mean µ = $\frac{1}{m}$
• standard deviation σ = µ
• percentile k : k = $\frac{ln\left(1-AreaToTheLeftOfk\right)}{\left(-m\right)}$
• P ( X > x ) = e (– mx )
• P ( a < X < b ) = e (– ma ) e (– mb )
• Memoryless Property: P ( X > x + k | X > x ) = P ( X > k )
• Poisson probability: with mean λ
• k ! = k *( k -1)*( k -2)*( k -3)…3*2*1

Data from the United States Census Bureau.

Data from World Earthquakes, 2013. Available online at http://www.world-earthquakes.com/ (accessed June 11, 2013).

“No-hitter.” Baseball-Reference.com, 2013. Available online at http://www.baseball-reference.com/bullpen/No-hitter (accessed June 11, 2013).

Zhou, Rick. “Exponential Distribution lecture slides.” Available online at www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf‎ (accessed June 11, 2013).

Use the following information to answer the next ten exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp (0.2)

What type of distribution is this?

Are outcomes equally likely in this distribution? Why or why not?

No, outcomes are not equally likely. In this distribution, more people require a little bit of time, and fewer people require a lot of time, so it is more likely that someone will require less time.

What is m ? What does it represent?

What is the mean?

five

What is the standard deviation?

State the probability density function.

f ( x ) = 0.2e -0.2 x

Graph the distribution.

Find P (2< x <10).

0.5350

Find P ( x >6).

Find the 70 th percentile.

6.02

Use the following information to answer the next seven exercises. A distribution is given as X ~ Exp (0.75).

What is m ?

What is the probability density function?

f ( x ) = 0.75 e -0.75 x

What is the cumulative distribution function?

Draw the distribution.

Find P ( x <4).

Find the 30 th percentile.

0.4756

Find the median.

Which is larger, the mean or the median?

The mean is larger. The mean is $\frac{1}{m}=\frac{1}{0.75}\approx 1.33$ , which is greater than 0.9242.

Use the following information to answer the next 16 exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

What is being measured here?

Are the data discrete or continuous?

continuous

In words, define the random variable X .

What is the decay rate ( m )?

m = 0.000121

The distribution for X is ______.

Find the amount (percent of one gram) of carbon-14 lasting less than 5,730 years. This means, find P ( x <5,730).

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x <5,730) = __________
1. Check student's solution
2. P ( x <5,730) = 0.5001

Find the percentage of carbon-14 lasting longer than 10,000 years.

1. Sketch the graph, and shade the area of interest.
2. Find the probability. P ( x >10,000) = ________

Thirty percent (30%) of carbon-14 will decay within how many years?

1. Sketch the graph, and shade the area of interest.
2. Find the value k such that P ( x < k ) = 0.30.
1. Check student's solution.
2. k = 2947.73

what are events in statistics
Like a roll of a dice! Or a coin toss. Or a gender reveal party!
what is statistics
can anyone explain it better for me
the science of statistics deal with the collection, analysis, interpretation and presentation of data
saquib
I am also studying statistics
saquib
Correlation regression, explain it to me in short.
guillio
correlation is used to find relationship between two and dependent ), regression used for predicting the future by analyzing past data
Arun
correlation is used to find relationship between two variables
Arun
dependent and independent eg. profit is dependent on sales
Arun
Statistics has been designed as the mathematical science of making decisions and drawing conclusions from data in situations of uncertainty. It includes the designings of experiments, collection, organization, summarization snd interpretation of numerical data.
Aliya
frequency distribution
noun STATISTICS a mathematical function showing the number of instances in which a variable takes each of its possible values.
Robin
ok
Common language-- taking a bunch of information and seeing if it is related or not to other info
Mandy
Does standard deviation have measuring unit?
Mohamed
yes, the measuring unit of the data you are looking at, for example centimetres for height.
Emma
thanks
Mohamed
is that easy to plot a graph between three axis?
Mohamed
yes we can but we do not have that much effective tools. If the graph is normal or less complicated then it is plotted effectively otherwise it will give you nightmare.
umair
whats the difference between discrete and contineous data
umar
Discrete variables are variables that can assume finite number of values. Continuous variables are variables that can assume infinite number of values
Mike
i will give you an example: {0,4,84} it contains discrete or limited values like it can also contain boolean values{true,false} or {0,1} and continuous are like {1,2,3,4,5......} , {0,0.1,0.2,0.3,0.4...........}
umair
a no. of values which are countable are called discrete variables on the other hand, a no. of values which are not countable are called continuous variables
Aliya
Yup, I would like to support Mr.Umair's argument by saying that it can only apply if we have a 3-D graph,otherwise a plane graph will not apply at all
festus
Aliya and Mike thnks to both of you ❤❤
umar
what's variance
what's case control study?
Shakilla
hi
Noman
?
Sulaiman
what is covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, (i.e., the variables tend to show simila
Robin
Economics department, faculty of social sciences, NOUN. You are required to calculate: the covariance and State whether the covariance is positive or negative. (11½ marks) Observation E D 1 15 17.24 2 16 15.00 3 8 14.91 4 6 4.50 5 15 18.00 6 12 6.29 7 12 19.23 8 18 18.69 9 12 7.21 10 20 4
Florence
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
Robin
what is the purpose of statistics and why it is important that statistics to be a solo and one complete field?
to organize,analyze and interpret information in order to make decision
Berema
what is noun?
so simple. the name of any person,place or thing.
Edu-info
Using the Chi-square test, two coins were flipped a hundred times. What will be the chances of getting a head and getting a tale? Given observed values is 62 heads and 38 tails. Expected value is 50 heads, 50 tails. Is the difference due to chance or a significant error? a. Draw your hypothesis
how can I win
what is difference between the blocking and confounding
how do you get 2/50 ?
can you explained it for me
korankye
an easier definition of inferential statistics
Inferential statistics makes inferences and predictions about a population based on a sample of data taken from the population in question.
Rukhsana
Inferential statistics helps you to extract insights from a random sample data which then helps you to use specific predictive Modeling/machine learning technic to predict or forecast.
Manish
what is stemplot? can anyone explain?
Javokhirbek
what is statistics
what is collection of data
ernest
no collection data was provided just the mean =14
Leticia
sd=14 describe the position of score to the mean how many points below or above z=1.00 z=1.50
Leticia
I have this sample score 14 18 12 22 14 22 21 20 13 26 13 26 16 21 they want me to.compute the z- score of x= 15 ×=40 and x=9?
Leticia
how do you understand that it is the mean?
Kenedy
fact and figure
hira
factors to consider when using secondary data
define binomial distribution
the distribution in which the outcome is of dichotomous
bimal
can you tell me Standar division is =14 what is the position of the score relative to the mean how many point above/below the mean?
Leticia By By By    By By By By Sam Luong  