2.9 – Example

Solution

Part 1

The given PDF must integrate to 1. Thus, we calculate

Therefore, k = 6/5.

Part 2

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

Part 3

The PDF and CDF of X are shown below.

Part 4

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.

Part 5

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):

Part 6

The mean time to complete a 1 hour exam is the expected value of the random variable X. Consequently, we calculate

Part 7

To find the variance of X, we use our alternate formula to calculate

Finally, we see that the standard deviation of X is