A student estimates that the amount of time needed to solve a homework problem is exponentially distributed and is independent of the amount of time needed to solve any other homework problem. On average it takes the student 10 minutes to solve a homework problem. Thus, if X is the amount of time (in minutes) it takes a student to solve a homework problem, the density of X is: f(x,λ) = λ e−λx for 0 ≤ x ≤ ∞, λ = 1 10 where λ is what R calls rate. Hint: This is a problem involving the exponential distribution. If you can determine the parameter λ for the distribution(given) then you should be able to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp(). a) What is the expected value of X? b) What is the variance of X? c) What is the standard deviation of X?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
A student estimates that the amount of time needed to solve a homework problem is exponentially distributed and is independent of the amount of time needed to solve any other homework problem. On average it takes the student 10 minutes to solve a homework problem. Thus, if X is the amount of time (in minutes) it takes a student to solve a homework problem, the density of X is:
for 0 ≤ x ≤ ∞, λ =
| 1 |
| 10 |
where λ is what R calls rate.
Hint: This is a problem involving the exponential distribution. If you can determine the parameter λ for the distribution(given) then you should be able to easily answer parts a ,b ,c and use the built-in R
a) What is the
b) What is the variance of X?
c) What is the standard deviation of X?
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 4 images
