2.9 – Example

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Problem

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,

  1. Find the value k that makes f(x) a probability density function (PDF)
  2. Find the cumulative distribution function (CDF)
  3. Graph the PDF and the CDF
  4. Use the CDF to find
    1. Pr(X ≤ 0)
    2. Pr(X ≤ 1)
    3. Pr(X ≤ 2)
  5. find the probability that that a randomly selected student will finish the exam in less than half an hour
  6. Find the mean time needed to complete a 1 hour exam
  7. Find the variance and standard deviation of X

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.

MATH105PDFCDF.jpg

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


source: http://wiki.ubc.ca/Science:MATH105_Probability/Lesson_2_CRV/2.12_Example

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