The time (sec) that it takes a librarian to locate an entry in a file of records on checked-out books has an exponential distribution with lambda symbol=0.5.
a)What proportion of all location times are less than 20 sec? At most 20 sec? Atleast 25 sec/ Between 10 sec and 25 sec?
I know the function is f(x)=0.5e^(-0.5x). If I take the integral of the function from 0 to 20, will that give me the answer. Because I got 1, so can you please check if 1 is the right answer for P(x<20).
Then less than 20 is: p(x<20)
betwen 10 and 25 is P(10<x<25)
how do I write at most 20 and at least 25?
You wrote:
How do I write at most 20 and at least 25?
At most 20:
P(x ≤ 20)
At least 25:
P(x ≥ 25)
Someone else may be able to help you with the function.
To find the proportions for different time intervals in an exponential distribution, you need to use the cumulative distribution function (CDF). The CDF gives you the probability that the random variable is less than or equal to a specific value.
For an exponential distribution with rate parameter λ, the CDF is given by:
CDF(x) = 1 - e^(-λx)
In this case, λ = 0.5, so the CDF becomes:
CDF(x) = 1 - e^(-0.5x)
a) Proportion of location times less than 20 seconds (P(x < 20)):
To find this proportion, you can simply evaluate the CDF at x = 20. So plug in 20 into the CDF equation:
P(x < 20) = CDF(20) = 1 - e^(-0.5 * 20)
Calculate this expression to find the answer.
b) Proportion of location times at most 20 seconds (P(x <= 20)):
To find this proportion, you need to subtract the probability of x > 20 from 1. So calculate:
P(x <= 20) = 1 - P(x > 20) = 1 - (1 - CDF(20))
c) Proportion of location times at least 25 seconds (P(x >= 25)):
To find this proportion, you need to subtract the probability of x < 25 from 1. So calculate:
P(x >= 25) = 1 - P(x < 25) = 1 - CDF(25)
d) Proportion of location times between 10 and 25 seconds (P(10 < x < 25)):
To find this proportion, you need to subtract the probability of x < 10 from the probability of x < 25. So calculate:
P(10 < x < 25) = P(x < 25) - P(x < 10) = CDF(25) - CDF(10)
By evaluating these expressions using the given CDF equation, you can find the proportions for each time interval.