For tasks 7-12, assume that the time interval between customers entering your st
ID: 3291657 • Letter: F
Question
For tasks 7-12, assume that the time interval between customers entering your store can be modeled using an exponential distribution. You know that you have an average of 4 customers per minute, so the rate is 4, or you can say that lambda = 4 according to Yakir (2011, p. 79-80).
It is easiest to keep everything in the original units of measurement (minutes), but you can also translate that to seconds because a rate of “4 customers per minute” is the same as “4 customer per 60 seconds,” and you can divide each number by 4 to get a rate of “1 customer per 15 seconds” or a rate of “1/15 customers per second.”
7) What is the expectation for the time interval between customers entering the store? Be sure to specify the units of measurement in your answer (see Yakir, 2011, pp. 79-80). Round to 3 decimal places: ___________________
8) What is the variance of the the time interval? Be sure to specify the units of measurement in your answer. Round to 3 decimal places:_________________
9) The pexp() function is introduced at the bottom of Yakir, 2011, p. 79, and there are some tips above. What is the probability that the time interval between customers entering the store will be less than 15.5 seconds. Be sure to enter values so that everything is in the same unit of measurement. Be sure to specify the units of measurement in your answer. Round your answer to 3 decimal places: _________________.
10) What is the probability that the time interval between customers entering the store will be between 10.7 seconds and 40.2 seconds (see Yakir (2011, p. 79-80)?________
11) The qexp() function in R allows you to enter a probability, and it will tell you the criterion value (“cutoff point”) that corresponds to that probability value (e.g., if you enter .05 it tells you the cutoff point below which 5% of the values in the distribution fall).
What value of x would be the criterion value (cut-off point) for the top 5% of time intervals (Round to 3 decimal places, and include the units of measurement)? _______
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12) Describe in your own words the meaning of the number that the following R command produces (you are asked to interpret the resulting number so that we understand what that number means).
pexp(1.2, rate=3)
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Information
You have now had practice with the binomial distribution and the exponential distribution. The approach to solving problems for the normal distribution is similar to that for the exponential function, but obviously you use different R functions (usually pnorm() or qnorm()).
For the following three exercises, assume that you have a normally distributed random variable, called A, with a mean of 7, and a population standard deviation of 3.
13) What is the probability that a randomly selected value from variable A will be greater than 9 (see Yakir, 2011 p. 88-89, 100)?_______
14) What value of variable A would be the cutoff point (criterion value) for identifying the lowest 4% of values in variable A (use the qnorm function)?____________
15) What is the probability that a randomly selected value from variable A will be more than one standard deviation above its mean (there are couple ways to solve this, one way is to use the standard normal distribution, Yakir, 2011, p. 90-91) ?________________
Book of Yakir can be reached at
http://pluto.huji.ac.il/~msby/StatThink/IntroStat.pdf
Explanation / Answer
time interval between customers entering your store can be modeled using an exponential distribution
average of 4 customers per minute, so the rate is 4, or you can say that lambda = 4
X is a continuous random variable since time is measured. It is given that = 4 minutes. To do any calculations, you must know m, the decay parameter.
m=1/
Therefore, m=1/4=0.25.
The standard deviation, , is the same as the mean. =
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 "ex." If you enter one for x, the calculator will display the value e.
The curve is:
f(x) = 0.25e–0.25x where x is at least zero and m = 0.25.
Poisson probability: P(X=k)= (kek)/k! with mean
a)time interval between customers entering the store = 0.25
b)variance of the the time interval = (Std. dev )2 = 2 (= 2) = 0.625
c)Pr(X<15.5) = 1 – e(–mx) =1 – e(–0.25*15.5) = 48.18
d)Pr(10.7X40.2)=Pr(X40.2)Pr(X10.7)= e^{-10.7/0.25} - e^{-40.2/0.25} = 0-0 = 0
e)qexp(Q,rate)
top 5% of time intervals so criterion point 1-0.05 =0.95
val of x = qexp(p, rate = 1, lower.tail = TRUE, log.p = FALSE)qexp(p, rate = 1, lower.tail = TRUE, log.p = FALSE)=
Pr[X>x]=1- e(–mx)=1-0.78 = 0.22
f) pExp(q, scale = 1, params = list(scale = 1), ...)
pexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE)
x, q
vector of quantiles.
p
vector of probabilities.
n
number of observations. If length(n) > 1, the length is taken to be the number required.
Rate vector of rates.
log, log.p logical; if TRUE, probabilities p are given as log(p). lower.tail logical; if TRUE (default), probabilities are Pr[Xx], otherwise, Pr[X>x]