The length of time X, needed by students in a particular course to complete a 1 hour exam is a random variable with PDF given by
For the random variable X,
- Find the value k that makes f(x) a probability density function (PDF)
- Find the cumulative distribution function (CDF)
- Graph the PDF and the CDF
- Use the CDF to find
- Pr(X ≤ 0)
- Pr(X ≤ 1)
- Pr(X ≤ 2)
- find the probability that that a randomly selected student will finish the exam in less than half an hour
- Find the mean time needed to complete a 1 hour exam
- Find the variance and standard deviation of X
The given PDF must integrate to 1. Thus, we calculate
Therefore, k = 6/5.
The CDF, F(x), is area function of the PDF, obtained by integrating the PDF from negative infinity to an arbitrary value x.
If x is in the interval (-∞, 0), then
If x is in the interval [0, 1], then
If x is in the interval (1, ∞) then
Note that the PDF f is equal to zero for x > 1. The CDF is therefore given by
The PDF and CDF of X are shown below.
These probabilities can be calculated using the CDF:
Note that we could have evaluated these probabilities by using the PDF only, integrating the PDF over the desired event.
The probability that a student will complete the exam in less than half an hour is Pr(X < 0.5). Note that since Pr(X = 0.5) = 0, since X is a continuous random variable, we an equivalently calculate Pr(x ≤ 0.5). This is now precisely F(0.5):
The mean time to complete a 1 hour exam is the expected value of the random variable X. Consequently, we calculate
To find the variance of X, we use our alternate formula to calculate
Finally, we see that the standard deviation of X is